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Dirac-type results for tilings and coverings in ordered graphs

Published online by Cambridge University Press:  21 November 2022

Andrea Freschi
Affiliation:
School of Mathematics, University of Birmingham, United Kingdom; E-mail: axf079@bham.ac.uk
Andrew Treglown
Affiliation:
School of Mathematics, University of Birmingham, United Kingdom; E-mail: a.c.treglown@bham.ac.uk

Abstract

A recent paper of Balogh, Li and Treglown [3] initiated the study of Dirac-type problems for ordered graphs. In this paper, we prove a number of results in this area. In particular, we determine asymptotically the minimum degree threshold for forcing

  1. (i) a perfect H-tiling in an ordered graph, for any fixed ordered graph H of interval chromatic number at least $3$ ;

  2. (ii) an H-tiling in an ordered graph G covering a fixed proportion of the vertices of G (for any fixed ordered graph H);

  3. (iii) an H-cover in an ordered graph (for any fixed ordered graph H).

The first two of these results resolve questions of Balogh, Li and Treglown, whilst (iii) resolves a question of Falgas-Ravry. Note that (i) combined with a result from [3] completely determines the asymptotic minimum degree threshold for forcing a perfect H-tiling. Additionally, we prove a result that, combined with a theorem of Balogh, Li and Treglown, asymptotically determines the minimum degree threshold for forcing an almost perfect H-tiling in an ordered graph (for any fixed ordered graph H). Our work therefore provides ordered graph analogues of the seminal tiling theorems of Kühn and Osthus [Combinatorica 2009] and of Komlós [Combinatorica 2000]. Each of our results exhibits some curious, and perhaps unexpected, behaviour. Our solution to (i) makes use of a novel absorbing argument.

Type
Discrete Mathematics
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

1 Introduction

In recent years there has been a significant effort to develop both Turán and Ramsey theories in the setting of vertex ordered graphs. A (vertex) ordered graph or labelled graph H on h vertices is a graph whose vertices have been labelled with $[h]:=\{1,\dots ,h\}$ . An ordered graph G with vertex set $[n]$ contains an ordered graph H on $[h]$ if (i) there is an injection $\phi : [h] \rightarrow [n]$ such that $\phi (i) < \phi (j)$ for all $1\leq i < j \leq h$ and (ii) $\phi (i)\phi (j)$ is an edge in G whenever $ij$ is an edge in H.

Turán-type problems concern edge density conditions that force a fixed graph H as a subgraph in a host graph G. Whilst the Erdős–Stone–Simonovits theorem [Reference Erdős and Simonovits8, Reference Erdős and Stone9] determines, up to a quadratic error term, the number of edges in the densest H-free n-vertex graph, there is still active interest in the Turán problem for bipartite H. Indeed, for bipartite H, the error term in the Erdős–Stone–Simonovits theorem is, in fact, the dominant term, so more refined results are sought. Similarly, a result of Pach and Tardos [Reference Pach and Tardos24] determines asymptotically the number of edges an ordered graph requires to force a copy of a fixed ordered graph H with so-called interval chromatic number $\chi _<(H)$ at least $3$ . Therefore, again there is significant interest in the ‘bipartite’ case of this problem (i.e., when $\chi _<(H)=2$ ); see Tardos [Reference Tardos, Lo, Mycroft, Perarnau and Treglown28] for a recent survey on such results.

The study of Ramsey theory for ordered graphs has also gained significant traction. For example, results of Conlon, Fox, Lee and Sudakov [Reference Conlon, Fox, Lee and Sudakov5], and of Balko, Cibulka, Král and Kynčl [Reference Balko, Cibulka, Král and Kynčl2], demonstrate that there are ordered graphs H for which the behaviour of the Ramsey number is vastly different to their underlying unordered graph.

Other than Turán and Ramsey problems, another central branch of extremal graph theory concerns Dirac-type results: that is, minimum degree conditions that force fixed (spanning) structures in graphs. In a recent paper, Balogh, Li and the second author [Reference Balogh, Li and Treglown3] initiated the study of Dirac-type results for ordered graphs. Their main focus was on perfect H-tilings, although they also raised other Dirac-type problems (see [Reference Balogh, Li and Treglown3, Section 8]). In both the ordered and unordered settings, an H-tiling in a graph G is a collection of vertex-disjoint copies of H contained in G. An H-tiling is perfect if it covers all the vertices of G. Perfect H-tilings are also often referred to as H-factors, perfect H-packings or perfect H-matchings. H-tilings can be viewed as generalisations of both the notion of a matching (which corresponds to the case when H is a single edge) and the Turán problem (i.e., a copy of H in G is simply an H-tiling of size one).

Balogh, Li and Treglown [Reference Balogh, Li and Treglown3] raised the following question.

Problem 1.1 [Reference Balogh, Li and Treglown3]

Given any ordered graph H and any $n \in \mathbb N$ divisible by $|H|$ , determine the smallest integer $\delta _{<}(H, n)$ such that every n-vertex ordered graph G with $\delta (G)\geq \delta _{<}(H, n)$ contains a perfect H-tiling.

The analogous problem in the (unordered) graph setting has been studied since the 1960s (see, e.g., [Reference Alon and Yuster1, Reference Corrádi and Hajnal6, Reference Hajnal and Szemerédi13, Reference Komlós, Sárkozÿ and Szemerédi17, Reference Kühn and Osthus20, Reference Kühn and Osthus21]), and 45 years later, a complete solution, up to an additive constant term, was obtained via a theorem of Kühn and Osthus [Reference Kühn and Osthus21]. We will discuss this result further when comparing this problem with Problem 1.1.

In [Reference Balogh, Li and Treglown3, Theorem 1.9], Balogh, Li and Treglown asymptotically resolved Problem 1.1 for H with $\chi _<(H)=2$ . Further, they developed approaches to the absorbing and regularity methods for ordered graphs, including providing general absorbing and almost perfect tiling lemmas. In this paper, we build on their results to asymptotically resolve Problem 1.1 in all remaining cases (i.e., for all H with interval chromatic number at least $3$ ). Our main result shows that Problem 1.1 does exhibit a somewhat different behaviour when $\chi _<(H)\geq 3$ compared to when $\chi _<(H) = 2$ ; we discuss this further in Section 1.3.

In addition to this result, we also resolve the Dirac-type problems for H-covers in ordered graphs (see Section 1.2) and H-tilings covering a fixed proportion x of the vertices of an ordered graph for $x \in (0,1)$ (see Section 1.4).

1.1 A Dirac-type theorem for perfect H-tilings

In this subsection, we state Theorem 1.8, which asymptotically resolves Problem 1.1 when $\chi _<(H)\geq 3$ . This result will depend on several definitions and parameters that we now introduce.

Definition 1.2 (Interval chromatic number)

Given $r \in \mathbb N$ , an interval r-colouring of an ordered graph H is a partition of the vertex set $[h]$ of H into r (possibly empty) intervals so that no two vertices belonging to the same interval are adjacent in H. The interval chromatic number $\chi _{<}(H)$ of an ordered graph H is the smallest $r\in {\mathbb N}$ such that there exists an interval r-colouring of H.

One can think of the interval chromatic number as the natural ordered analogue of the chromatic number of a graph. Moreover, whilst the Erdős–Stone–Simonovits theorem [Reference Erdős and Simonovits8, Reference Erdős and Stone9] asserts that $1-1/(\chi (H)-1)+o(1)$ is the edge density threshold for ensuring a copy of a graph H in G, the Erdős–Stone–Simonovits theorem for ordered graphs due to Pach and Tardos [Reference Pach and Tardos24] asserts the corresponding threshold in the ordered graph setting is $1-1/(\chi _< (H)-1)+o(1)$ .

The next definition is the relevant parameter for studying almost perfect H-tilings in graphs.

Definition 1.3 (Critical chromatic number)

The critical chromatic number $\chi _{cr}(F)$ of an unordered graph F is defined as

$$\begin{align*}\chi_{cr}(F):=(\chi(F) - 1)\frac{|F|}{|F| - \sigma(F)}, \end{align*}$$

where $\sigma (F)$ denotes the size of the smallest possible colour class in any $\chi (F)$ -colouring of F.

Note that $\chi (H)-1<\chi _{cr}(H)\leq \chi (H)$ for all graphs H, and $\chi _{cr}(H)=\chi (H)$ precisely when every $\chi (H)$ -colouring of H has colour classes of equal size.

We informally refer to an H-tiling in an n-vertex (ordered) graph G as an almost perfect H-tiling if it covers all but at most $o(n)$ vertices of G. Komlós [Reference Komlós16] proved that $(1-1/\chi _{cr}(H))n$ is the minimum degree threshold for forcing an almost perfect H-tiling in an n-vertex graph G. In fact, it was later shown [Reference Shokoufandeh and Zhao26] that such graphs G contain H-tilings covering all but a constant number of the vertices in G. In the setting of ordered graphs, a related parameter $\chi ^*_{cr}(H)$ turns out to be the relevant parameter for forcing an almost perfect H-tiling. To introduce this parameter, we need the following definitions.

For two subsets $X, Y$ of $[n]$ , we write $X<Y$ if $x<y$ for all $x\in X$ and $y\in Y$ . Let B be a complete k-partite unordered graph with parts $U_1,\ldots , U_k$ and $\sigma $ be a permutation of the set $[k]$ . An interval labelling of B with respect to $\sigma $ is a bijection $\phi : V(B)\rightarrow [|B|]$ such that $\phi (U_i)<\phi (U_j)$ if $\sigma (i)<\sigma (j)$ : that is,

$$ \begin{align*} \phi(U_{\sigma^{-1}(1)})<\cdots<\phi(U_{\sigma^{-1}(k)}). \end{align*} $$

For brevity, we will usually drop $\phi $ and just write $U_{\sigma ^{-1}(1)}<\cdots <U_{\sigma ^{-1}(k)}.$ Given $t \in \mathbb N$ , write $B(t)$ for the blow-up of B with vertex set $\bigcup _{x\in V(B)}V_x$ , where the $V_x$ s are sets of t independent vertices; so there are all possible edges between $V_x$ and $V_y$ in $B(t)$ if $xy \in E(B)$ . Given an interval labelling $\phi $ of B, let $(B(t), \phi )$ be the ordered graph obtained from $B(t)$ by equipping $V(B(t))$ with a vertex ordering, satisfying $V_x < V_y$ for every $x, y\in V(B)$ with $\phi (x)<\phi (y)$ . We refer to $(B(t), \phi )$ as an ordered blow-up of B.

Definition 1.4 (Bottlegraph)

For an ordered graph H, we say that a complete k-partite unordered graph B is a bottlegraph of H if, for every permutation $\sigma $ of $[k]$ and every interval labelling $\phi $ of B with respect to $\sigma $ , there exists a constant $t=t(B, H, \phi )$ such that the ordered blow-up $(B(t), \phi )$ contains a perfect H-tiling. We say that B is a simple bottlegraph of H if, for any choice of $\sigma $ and $\phi $ , we can take $t=1$ .

Note that in Definition 1.4, we did not impose any restriction on the size of the parts of a bottlegraph. However, as we will see in Proposition 5.1, it suffices to consider bottlegraphs $B'$ , where all parts are of the same size except for perhaps one smaller part. More precisely, given any bottlegraph B of H, there is another bottlegraph $B'$ with this structure such that $\chi _{cr}(B')=\chi _{cr}(B)$ . This bottle-like structure is where the name bottlegraph is derived and was first used by Komlós [Reference Komlós16] in the setting of unordered graphs.

Definition 1.5 (Ordered critical chromatic number)

The ordered critical chromatic number $\chi ^*_{cr}(F)$ of an ordered graph F is defined as

$$ \begin{align*} \chi_{cr}^*(F):=\inf\{\chi_{cr}(B):B\text{ is a bottlegraph of }F\}. \end{align*} $$

We say a bottlegraph B of F is optimal if $\chi _{cr}(B)=\chi _{cr}^*(F)$ .

Notice that $\chi _<(H)-1 \leq \chi _{cr}^*(H)$ for all ordered graphs H as each bottlegraph B of H must have chromatic number at least $\chi _<(H)$ so $\chi _<(H)-1 < \chi _{cr}(B)$ . In fact, Proposition 2.6 in Section 2 yields a stronger lower bound on $\chi _{cr}^*(H)$ . On the other hand, (in contrast to $\chi _{cr}(F)$ for unordered graphs F), we will also see examples of ordered graphs where $\chi _{cr}^*(H)$ is much larger than $\chi _<(H)$ . Note though that $\chi _{cr}^*(H)\leq h$ for any ordered graph H on $[h]$ as $K_{h}$ is a bottlegraph of H. In fact, this upper bound is attained when H is such that $1$ and $2$ are adjacent or $h-1$ and h are adjacent; this is an immediate consequence of Proposition 11.1. To aid the reader’s intuition, in Section 3.3, we give examples of ordered graphs H, where we compute $\chi ^*_{cr}(H)$ . Various bounds on $\chi ^*_{cr}(H)$ are given in Section 11.

The next result, a simple corollary of [Reference Balogh, Li and Treglown3, Theorem 4.3], shows that $\chi ^*_{cr}(H)$ is a relevant parameter for forcing an almost perfect H-tiling in an ordered graph.Footnote 1

Theorem 1.6 (Balogh, Li and Treglown [Reference Balogh, Li and Treglown3])

Let H be an ordered graph. Then for every $\eta>0$ , there exists an integer $n_0=n_0(H,\eta )$ so that every ordered graph G on $n\geq n_0$ vertices with

$$ \begin{align*} \delta(G)\geq\left(1-\frac{1}{\chi^*_{cr}(H)}\right)n \end{align*} $$

contains an H-tiling covering all but at most $\eta n$ vertices.

At first sight, it is not clear if the minimum degree threshold in Theorem 1.6 is best possible. However, Theorem 12.1 in Section 12 shows that Theorem 1.6 is best possible in the sense that one cannot replace the $1-{1}/{\chi ^*_{cr}(H)}$ term in the minimum degree condition with any other fixed constant term $a< 1-{1}/{\chi ^*_{cr}(H)}$ . Thus, Theorem 12.1 and Theorem 1.6 provide an analogue of Komlós’ theorem in the ordered setting.

Unusually, in the proof of Theorem 12.1, for most ordered graphs H, we do not simply produce an explicit extremal example. Indeed, if one has not explicitly computed the value of $\chi ^*_{cr}(H)$ and the ‘reason’ it takes this value, then it seems difficult to produce such an explicit extremal example. Instead, the proof splits into a few cases and uses various tools and results that we introduce in the paper.

At this point, the reader may wonder if the conclusion of Theorem 1.6 can be strengthened to ensure a perfect H-tiling. For some ordered graphs H, this is possible. However, for other ordered graphs, one will require a significantly higher minimum degree condition. The following definition is the critical concept for articulating this dichotomy for H with $\chi _<(H) \geq 3$ .

Definition 1.7 (Local barrier)

Let H be an ordered graph on $[h]$ with $r:=\chi _<(H)\geq 2$ . We say that H has a local barrier if for some fixed $i\ne j\in [r+1]$ , the following condition holds. Given any interval $(r+1)$ -colouring of H with colour classes $V_1<\cdots <V_{r+1}$ such that $V_i=\{v\}$ is a singleton class, there is at least one edge between v and $V_j$ in H.

Note that in this definition, we may have that a colour class $V_k$ is empty. If H is the ordered complete graph on r vertices, then H does not have a local barrier; it is also easy to check that $\chi ^*_{cr}(H)=\chi _<(H)=r$ . Given $r \geq 2,$ let $H'$ be any complete r-partite (unordered) graph with at least $2$ vertices in each colour class. Let H be any ordered graph obtained from $H'$ by assigning an interval labelling to $H'$ ; so $\chi _<(H)=r$ . Then one can check that H has a local barrier with parameters $i=1$ and $j=r+1$ as in Definition 1.7.

We are now able to state our main result, which resolves Problem 1.1 for all ordered graphs H with $\chi _<(H) \geq 3$ .

Theorem 1.8. Let H be an ordered graph with $\chi _<(H)\geq 3$ . Given any $\eta>0$ , there exists an integer $n_0=n_0(H,\eta )$ so that if $n\geq n_0$ and $|H|$ divides n, then

  1. (i) $ \left (1-\frac {1}{\chi ^*_{cr}(H)} \right )n \leq \delta _<(H,n)\leq \left (1-\frac {1}{\chi ^*_{cr}(H)} +\eta \right )n $ if $\chi ^*_{cr}(H) \geq \chi _<(H)$ ;

  2. (ii) $ \left (1-\frac {1}{\chi _<(H)} \right )n< \delta _<(H,n)\leq \left (1-\frac {1}{\chi _<(H)} +\eta \right )n $ if $\chi ^*_{cr}(H) < \chi _<(H)$ and H has a local barrier;

  3. (iii) $ \left (1-\frac {1}{\chi ^*_{cr}(H)} \right )n \leq \delta _<(H,n)\leq \left (1-\frac {1}{\chi ^*_{cr}(H)} +\eta \right )n $ if $\chi ^*_{cr}(H) < \chi _<(H)$ and H has no local barrier.

Therefore Problem 1.1 is now asymptotically resolved. The reader might find it hard to see why the value of $\delta _<(H,n)$ behaves as in Theorem 1.8, and indeed, it took the authors quite some time to discover the correct behaviour of this problem. In Section 1.3, we give further intuition on this result. In Section 3, we give examples of H in each case (i)–(iii) of the theorem. In Section 2, we give the extremal constructions for Theorem 1.8. In particular, in cases (i) and (iii), the extremal examples are ‘bottlegraphs’ – complete multipartite ordered graphs where each part has the same size except at most one smaller part; in Section 11, we explicitly compute the value of $\chi ^*_{cr}(H)$ for a range of H and thus the minimum degree threshold in Theorem 1.8 also. The explanation for why these extremal ‘bottlegraphs’ do not have perfect H-tilings revolves around ‘space barrier’ constraints (i.e., one runs out of space in some subset of vertices in the extremal bottlegraph).Footnote 2 However, the ‘reason’ for obtaining a space barrier can be somewhat involved (and unlike any other space barrier extremal example we have ever seen before); we discuss this in Section 3.1.

The proof of Theorem 1.8 applies an absorbing theorem from [Reference Balogh, Li and Treglown3, Theorem 4.1] and Theorem 1.6 above. The main novelty is to prove an absorbing theorem for ordered graphs H as in Theorem 1.8(iii). Whilst our argument makes use of a lemma of Lo and Markström [Reference Lo and Markström23] and seems rather natural, it is different to any absorbing proof we have previously seen (in particular, we do not use local-global absorbing as in [Reference Balogh, Li and Treglown3]). See Section 9 for an overview of our absorbing strategy.

1.2 A Dirac-type theorem for vertex covers

Given (ordered) graphs H and G, we say that G has an H-cover if every vertex in G lies in a copy of H. Note that the notion of an H-cover is an ‘intermediate’ between seeking a single copy of H and a perfect H-tiling; in particular, a perfect H-tiling in G is itself an H-cover. Given any $n \in \mathbb N$ and any (ordered) graph H, let $\delta _{\text {cov}}(H,n)$ denote the smallest integer k such that every n-vertex (ordered) graph G with $\delta (G)\geq k$ contains an H-cover. As noted in [Reference Kühn, Osthus and Treglown22], an easy application of Szemerédi’s regularity lemma asymptotically determines $\delta _{\text {cov}}(H,n)$ for all graphs H.

Proposition 1.9 [Reference Kühn, Osthus and Treglown22, Proposition 6]

For every graph H and every $\eta>0$ , there exists an integer $n_0=n_0(H,\eta )$ so that if $n \geq n_0$ , then

$$ \begin{align*} \left (1-\frac{1}{\chi(H)-1} \right )n-1 \leq \delta_{\text{cov}}(H,n)\leq \left (1-\frac{1}{\chi(H)-1} +\eta \right )n. \end{align*} $$

Proposition 1.9 implies that, asymptotically, the minimum degree threshold for ensuring an H-cover in a graph G is the same as the minimum degree threshold for ensuring a single copy of H in G. In [Reference Kühn, Osthus and Treglown22, Theorem 5], Kühn, Osthus and Treglown asymptotically determined the Ore-type degree condition that forces an H-cover for any fixed graph H. There have also been several recent papers concerning minimum $\ell $ -degree conditions that force H-covers in k-uniform hypergraphs; see, for example, [Reference Falgas-Ravry, Markström and Zhao11, Reference Falgas-Ravry and Zhao12, Reference Han, Lo and Sanhueza-Matamala14].

Falgas-Ravry [Reference Falgas-Ravry10] raised the question of determining $\delta _{\text {cov}}(H,n)$ for all ordered graphs H. Our next result asymptotically answers this question.

Theorem 1.10. Let H be an ordered graph and $\eta>0$ . Then there exists an integer $n_0=n_0(H,\eta )$ so that if $n \geq n_0$ , then

  1. (i) $ \left (1-\frac {1}{\chi _<(H)-1} \right )n < \delta _{\text {cov}}(H,n)\leq \left (1-\frac {1}{\chi _<(H)-1} +\eta \right )n $ if H has no local barrier;

  2. (ii) $ \left (1-\frac {1}{\chi _<(H)} \right )n <\delta _{\text {cov}}(H,n)\leq \left (1-\frac {1}{\chi _<(H)} +\eta \right )n $ if H has a local barrier.

Theorem 1.10 is a direct consequence of some of the auxiliary results we use in the proof of Theorem 1.8. Note that the behaviour of the threshold in Theorem 1.10 is perhaps unexpected. Indeed, unlike in the unordered setting, Theorem 1.10 and the Erdős–Stone–Simonovits theorem for ordered graphs imply that the asymptotic minimum degree thresholds for forcing a copy of H and an H-cover are different if H has a local barrier.

Furthermore, a key moral of the Erdős–Stone–Simonovits theorem (and Proposition 1.9) is that once a graph G is dense enough (or has large enough minimum degree) to ensure a copy of $K_r$ (or a $K_r$ -cover), then G must contain every fixed graph H (or an H-cover) for every H of chromatic number r. An intuition for this comes from Szemerédi’s regularity lemma. However, the analogous moral is not true for H-covers in ordered graphs. Indeed, if H is an ordered complete graph on r vertices (so $\chi _<(H)=r$ and H has no local barrier), then Theorem 1.10 tells us the minimum degree threshold for forcing an H-cover in an n-vertex ordered graph is $(1-\frac {1}{r-1}+o(1) )n$ whilst the corresponding threshold for any ‘blow-up’ $H'$ of H (so $\chi _<(H')=\chi _<(H)=r$ and H has a local barrier) is significantly higher, namely $(1-\frac {1}{r}+o(1) )n$ . This should hint to the reader that the regularity method behaves differently in the ordered setting; in particular, if H has a local barrier, this provides an obstruction when applying the regularity lemma. More discussion on the regularity method for ordered graphs can be found in [Reference Balogh, Li and Treglown3, Section 3.1].

1.3 Intuition behind the threshold in Theorem 1.8

In this subsection, we build up further intuition behind the threshold in Theorem 1.8. For this, it will be useful to first take a step back and consider perfect H-tilings in unordered graphs. In this setting, the Dirac-type threshold is governed by two factors:

  • $(C1)$ The minimum degree needs to be large enough to force an almost perfect H-tiling.

  • $(C2)$ The minimum degree must be large enough to prevent ‘divisibility’ barriers within the host graph that constrain us from turning an almost perfect H-tiling into a perfect H-tiling.

This is made precise by the following theorem of Kühn and Osthus [Reference Kühn and Osthus21]. (See [Reference Kühn and Osthus21, Section 1.2] for the definition of $\text {hcf}(H)=1$ .)

Theorem 1.11 (Kühn and Osthus [Reference Kühn and Osthus21])

Let $\delta (H, n)$ denote the smallest integer k such that every graph G whose order n is divisible by $|H|$ and with $\delta (G)\geq k$ contains a perfect H-tiling. For every unordered graph H,

$$\begin{align*}\delta(H, n)=\left(1 - \frac{1}{\chi^*(H)}\right)n +O(1), \end{align*}$$

where $\chi ^*(H):=\chi _{cr} (H)$ if $\text {hcf}(H)=1$ and $\chi ^*(H):=\chi (H)$ otherwise.

Recall that every graph H satisfies $\chi _{cr} (H)\le \chi (H)$ . So by Komlós’ aforementioned almost perfect tiling theorem [Reference Komlós16], the minimum degree condition in Theorem 1.11 is enough to ensure that $(C1)$ holds. Meanwhile, those graphs H with $\text {hcf}(H)=1$ are precisely the graphs for which, at the almost perfect tiling threshold, $(C2)$ is satisfied. Furthermore, for graphs H with $\text {hcf}(H)\ne 1$ , $(C2)$ is only guaranteed to be satisfied once the n-vertex host graph has minimum degree around $(1-1/\chi (H))n$ .

We do not state the precise definition of $\text {hcf}(H)=1$ here; however, the following example is instructive. Let H be any connected bipartite graph, and let $n \in \mathbb N $ be divisible by $|H|$ . Consider the n-vertex graph G that consists of two disjoint cliques whose sizes are as equal as possible so that neither is divisible by $|H|$ . Then whilst G contains an almost perfect H-tiling, the divisibility constraint on the clique sizes prevents a perfect H-tiling. Thus all such H are examples of graphs with $\text {hcf}(H) \ne 1$ . In particular, $\delta (G)=n/2-O(1)=(1-1/\chi (H))n-O(1)$ , so G is an extremal example for Theorem 1.11 in this case.

As mentioned earlier, another necessary condition for a Dirac-type threshold for perfect H-tilings is the following:

  • $(C3)$ The minimum degree needs to be large enough to force an H-cover.

Condition $(C3)$ , however, does not factor into the statement of Theorem 1.11 as Proposition 1.9 shows that one can ensure an H-cover ‘earlier’ than an almost perfect H-tiling (recall that $\chi (H)-1< \chi _{cr}(H)$ ).

Interestingly, the opposite is true in the ordered graph setting when H has a local barrier and $\chi _<(H)\geq 3$ is such that $\chi _<(H)> \chi _{cr}^* (H)$ . Indeed, in this case, Theorem 1.8(ii) essentially states that the H-cover condition is the ‘last’ of conditions $(C1)$ $(C3)$ to be satisfied. In particular, Extremal Example 1 in Section 2 shows that for every H satisfying Definition 1.7, there are n-vertex ordered graphs with $\delta (G)> (1-1/\chi _<(H))n-1$ for which a certain vertex does not lie in a copy of H.

In all other cases when $\chi _<(H)\geq 3$ , Theorem 1.8 essentially states that the almost perfect tiling condition is the ‘last’ of conditions $(C1)$ $(C3)$ to be satisfied. Therefore, surprisingly (at least to the authors!), divisibility barriers play no role in Problem 1.1 for H with $\chi _<(H)\geq 3$ . In contrast, Theorem 1.9 in [Reference Balogh, Li and Treglown3] shows that in the case when $\chi _<(H)=2$ , each of $(C1)$ , $(C2)$ and $(C3)$ can be the condition that governs the Dirac-type threshold for perfect H-tiling, depending on the choice of H.

1.4 A Dirac-type theorem for H-tilings

In addition to determining the Dirac-type threshold for almost perfect tilings in unordered graphs, Komlós [Reference Komlós16] provided a best-possible minimum degree condition for forcing an H-tiling covering a certain proportion of the vertices in a graph G.

Definition 1.12 ( $(x,H)$ -tilings)

Let G and H be (ordered) graphs and $x\in [0,1]$ . An $(x,H)$ -tiling in G is an H-tiling covering at least $x|G|$ vertices. So a $(1,H)$ -tiling is simply a perfect H-tiling.

Theorem 1.13 (Komlós [Reference Komlós16])

Let H be a graph and $x\in (0,1)$ . Define

$$ \begin{align*} g(x,H):=\left(1-\frac{1}{\chi(H)-1}\right)(1-x)+\left(1-\frac{1}{\chi_{cr}(H)}\right)x. \end{align*} $$

Given any $\eta>0$ , there exists some $n_0=n_0(x,H,\eta )\in \mathbb {N}$ such that if G is a graph on n vertices where $n\geq n_0$ and $\delta (G)\geq g(x,H)\cdot n$ , then there exists an $(x-\eta ,H)$ -tiling in G.

Note that the minimum degree condition in Theorem 1.13 is the best possible in the sense that given any fixed H and x ∈ (0, 1), one cannot replace g(x, H) with any fixed g′(x, H) < g(x, H) (see [Reference Komlós16, Theorem 7] for a proof of this).

The function $g(x,H)$ is quite well-behaved. Indeed, for fixed H, $g(x,H)$ grows linearly in x. Note that $g(0,H)\cdot n$ and $g(1,H)\cdot n$ are the asymptotic minimum degree thresholds for ensuring an n-vertex graph contains a copy of H and an almost perfect H-tiling, respectively. From this perspective, the function $g(x,H)$ can be viewed as a linear interpolation of these two thresholds.

The question of obtaining an ordered graph analogue of Theorem 1.13 was raised in [Reference Balogh, Li and Treglown3, Question 8.2]. We provide an answer to this problem; for this, we require the following definitions.

Definition 1.14 (x-bottlegraphs)

Let H be an ordered graph and $x\in (0,1]$ . An unordered graph B is an x-bottlegraph of H if it satisfies the following properties:

  1. (i) B is a complete k-partite graph with parts $U_1,U_2,\dots ,U_k$ for some $k\in \mathbb {N}$ .

  2. (ii) There exists some $m\in \mathbb {N}$ such that $|U_1|\leq m$ and $|U_i|=m$ for every $i>1$ .

  3. (iii) Given any permutation $\sigma $ of $[k]$ and any interval labelling of B with respect to $\sigma $ , the resulting ordered graph contains an $(x,H)$ -tiling.

Definition 1.15. Let H be an ordered graph and $x\in (0,1]$ . We define $\chi _{cr}^*(x,H)$ as

$$ \begin{align*} \chi_{cr}^*(x,H):=\inf \{\chi_{cr}(B) \, : \, B \, \text{is an } x\text{-bottlegraph of } H\}. \end{align*} $$

Given any ordered graph H, if B is an x-bottlegraph of H, then $\chi (B) \geq \chi _<(H)$ . This implies that $\chi _{cr}(B)> \chi _<(H)-1$ , so

(1) $$ \begin{align} \chi ^* _{cr}(x,H) \geq \chi _< (H)-1. \end{align} $$

An application of Theorem 1.13 together with a tool from [Reference Balogh, Li and Treglown3, Lemma 6.2] yields the following minimum degree condition for the existence of $(x,H)$ -tilings in ordered graphs.

Theorem 1.16. Let H be an ordered graph and $x\in (0,1)$ , and define

$$ \begin{align*} f(x,H):=\left(1-\frac{1}{\chi_{cr}^*(x,H)}\right). \end{align*} $$

Given any $\eta>0$ , there exists some $n_0=n_0(x,H,\eta )\in \mathbb {N}$ such that if G is an ordered graph on n vertices with $n\geq n_0$ and $\delta (G)\geq (f(x,H)+\eta ) n$ , then G contains an $(x,H)$ -tiling.

The minimum degree condition in Theorem 1.16 is the best possible in the following sense. Let H and $x\in (0,1)$ be fixed. Given any $0<a <1-1/\chi _{cr}^*(x,H)$ and any sufficiently large $n \in \mathbb N$ , consider any n-vertex graph B that satisfies (i) and (ii) of Definition 1.14 for some choice of $k,m \in \mathbb N$ and where

$$ \begin{align*} an < \delta (B) = n-m= \left(1-\frac{1}{\chi_{cr}(B)}\right)n< f(x,H)\cdot n. \end{align*} $$

So $\chi _{cr}(B) <\chi _{cr}^*(x,H)$ . (Note such a graph B exists for any choice of $0<a <1-1/\chi _{cr}^*(x,H)$ .) Then by definition of $\chi _{cr}^*(x,H)$ , there is a permutation $\sigma $ of $[k]$ and an interval labelling $\phi $ of B with respect to $\sigma $ such the resulting ordered graph $(B,\phi )$ does not contain an $(x,H)$ -tiling.

A drawback of Theorem 1.16 is that it seems hard to compute $\chi _{cr}^*(x,H)$ in general. However, in Section 14, we describe the behaviour of the function $f(x,H)$ for some fixed ordered graphs H. In particular, akin to Theorem 1.13, if H has $\chi _<(H)=2$ , then $f(x,H)$ is linear in x. Perhaps surprisingly, though, there are ordered graphs where $f(x,H)$ is only piecewise linear. We also compute $f(x,H)$ for every ordered graph H and every x that is not too big.

1.5 Organisation of the paper

The paper is organised as follows. In Section 2, we give the extremal constructions for Theorems 1.8 and 1.10. In Section 3, we give some examples of ordered graphs H that fall into each of the three cases of Theorem 1.8. In Section 4, we state a new absorbing theorem (Theorem 4.2) and an absorbing theorem from [Reference Balogh, Li and Treglown3] and combine them with Theorem 1.6 to prove Theorem 1.8. The subsequent sections therefore build up tools for the proof of Theorem 4.2: in Section 5, we state a couple of useful properties of bottlegraphs; in Section 6, we introduce Szemerédi’s regularity lemma and related useful results; some tools for absorbing are given in Section 7; Section 8 contains several results that give flexibility in how one can interval colour certain ordered graphs H.

In Section 9, we give a sketch of the proof of Theorem 4.2 before proving it and Theorem 1.10 in Section 10. In Section 11, we give general upper and lower bounds on $\chi ^*_{cr}(H)$ and also compute $\chi ^*_{cr}(H)$ for a few general classes of ordered graphs H.

In Section 12, we prove that the minimum degree condition in Theorem 1.6 is the best possible. The proof of Theorem 1.16 is given in Section 13; in the subsequent section, we describe the behaviour of the function $f(x,H)$ for some choices of H. We conclude the paper with some open problems in Section 15.

1.6 Notation

Given $n\in \mathbb N$ , let $[n]:=\{1, \ldots , n\}$ . A nearly balanced interval partition of $[n]$ is a partition of $[n]$ into intervals $W_1<\dots <W_t$ , where $||W_i|-|W_j||\leq 1$ for every $1\leq i, j \leq t$ . Similarly, a t-partite graph with vertex classes $V_1,\ldots , V_t$ is nearly balanced if $||V_i|-|V_j||\leq 1$ for every $1\leq i, j \leq t$ .

If G is an (ordered) graph, $|G|$ denotes the size of its vertex set, and $e(G)$ denotes the number of edges in G. Given $A\subseteq V (G)$ , the induced subgraph $G[A]$ is the subgraph of G whose vertex set is A and whose edge set consists of all of the edges of G with both endpoints in A. We define $G\setminus A:=G[V(G)\setminus A]$ . For two disjoint subsets $A, B\subseteq V (G)$ , the induced bipartite subgraph $G[A, B]$ is the subgraph of G whose vertex set is $A\cup B$ and whose edge set consists of all of the edges of G with one endpoint in A and the other endpoint in B. We write $e(A,B):=e(G[A,B])$ .

For an (ordered) graph G and a vertex $x \in V(G)$ , we define $N_G(x)$ as the set of neighbours of x in G and $d_G (x):=|N_G(x) |$ . For $X \subseteq V(G)$ , we define $d_G (x,X):=|N_G(x)\cap X |$ . Given (ordered) graphs G and H and $X \subseteq V(G)$ , we say that $G[X]$ spans a copy of H in G if H is a spanning subgraph of $G[X]$ .

Given an ordered graph G, we say that $V_1<\dots <V_r$ is an interval r-colouring of G to mean there is an interval r-colouring of G with colour classes $V_1< \dots <V_r$ . We say that an ordered graph G is complete r-partite if there exists an interval r-colouring $V_1<\cdots <V_r$ such that $xy\in E(G)$ for every $x\in V_i$ and $y\in V_j$ with $i\not =j$ . We refer to the $V_i$ s as the parts of G.

Given an unordered graph G and a positive integer t, let $G(t)$ be the graph obtained from G by replacing every vertex $x\in V(G)$ by a set $V_x$ of t vertices spanning an independent set and joining $u\in V_x$ to $v\in V_y$ precisely when $xy$ is an edge in G; that is, we replace the edges of G by copies of $K_{t,t}$ . We will refer to $G(t)$ as a blown-up copy of G. If $U_i$ is a vertex class in G, then we write $U_i(t)$ for the corresponding vertex class in $G(t)$ . We use analogous notation when considering blown-up copies of complete k-partite ordered graphs. In particular, given a complete k-partite ordered graph B with parts $B_1<\dots < B_k$ , the ordered blow-up $B(t)$ of B consists of parts $B_1(t)<\dots < B_k(t)$ , where $|B_i(t)|=t|B_i|$ for all $i \in [k]$ .

Throughout the paper, we omit all floor and ceiling signs whenever they are not crucial. The constants in the hierarchies used to state our results are chosen from right to left. For example, if we claim that a result holds whenever $0< a\ll b\ll c\le 1$ , then there are nondecreasing functions $f:(0,1]\to (0,1]$ and $g:(0,1]\to (0,1]$ such that the result holds for all $0<a,b,c\le 1$ with $b\le f(c)$ and $a\le g(b)$ . Note that $a \ll b$ implies that we may assume in the proof that, for example, $a < b$ or $a < b^2$ .

2 Extremal constructions

In this section, we provide the extremal examples for Theorems 1.8 and 1.10. First, consider the case when H has a local barrier. We now construct an n-vertex ordered graph that does not contain an H-cover (and thus no perfect H-tiling) and whose minimum degree is more than $(1-1/\chi _<(H))n-1$ , thereby giving the lower bounds in Theorem 1.8(ii) and Theorem 1.10(ii).

Extremal Example 1. Let $n,r\in \mathbb {N}$ and $i,j\in [r+1]$ with $i\not =j$ . Let $F_1(n,r,i,j)$ be an n-vertex ordered graph consisting of vertex classes $U_1<\cdots <U_{r+1}$ that satisfy the following conditions:

  • $U_i=\{u\}$ is a singleton class while the remaining vertex classes are as equally sized as possible, and in particular, $|U_j|=\left \lfloor \frac {n-1}{r}\right \rfloor $ ;

  • $F_1(n,r,i,j) \setminus \{u\}$ is a complete r-partite ordered graph with parts $U_1,\dots , U_{i-1}, U_{i+1}, \dots U_{r+1}$ ;

  • u is adjacent to all other vertices except those in $U_j$ .

Note that

(2) $$ \begin{align} \delta(F_1(n,r,i,j))=n-1-|U_j|=n-1-\left\lfloor\frac{n-1}{r}\right\rfloor>\left(1-\frac{1}{r}\right)n-1. \end{align} $$

Furthermore, we now prove that $F_1(n,r,i,j)$ does not contain an H-cover (or a perfect H-tiling), provided that $\chi _<(H)=r$ and H has a local barrier with respect to parameters $i,j\in [r+1]$ .

Lemma 2.1. Let H be an ordered graph, let $r:=\chi _<(H)$ , and let $n\in \mathbb {N}$ . If H has a local barrier, then there exist $i,j\in \mathbb {N}$ , with $i\not =j$ and a vertex $u\in F_1(n,r,i,j)$ , such that there is no copy of H in $F_1(n,r,i,j)$ covering the vertex u. In particular, $F_1(n,r,i,j)$ does not contain an H-cover or a perfect H-tiling.

Proof. Suppose H has a local barrier with respect to $i\ne j\in [r+1]$ , as defined in Definition 1.7. Let u be the vertex in the singleton class $U_i$ of $F_1(n,r,i,j)$ . Suppose there is a copy of H in $F_1(n,r,i,j)$ covering the vertex u. Then the interval $(r+1)$ -colouring $U_1<\cdots <U_{r+1}$ of $F_1(n,r,i,j)$ induces an interval $(r+1)$ -colouring $V_1<\cdots <V_{r+1}$ of H such that $V_i=\{v\}$ is a singleton class and there is no edge between v and $V_j$ . This contradicts the assumption that H has a local barrier with respect to $i,j$ ; thus, there is no copy of H in $F_1(n,r,i,j)$ covering the vertex u.

We immediately obtain the following corollary of Lemma 2.1 and (2).

Corollary 2.2. Let H be an ordered graph, and let $n\in \mathbb N$ . If H has a local barrier, then

$$ \begin{align*} \delta_{cov}(H,n)>\left(1-\frac{1}{\chi_<(H)}\right)n \end{align*} $$

and, if $|H|$ divides n

$$ \begin{align*} \delta_<(H,n)>\left(1-\frac{1}{\chi_<(H)}\right)n. \end{align*} $$

Next we prove a general lower bound on $\delta _<(H,n)$ , which is asymptotically sharp if the ordered graph H does not have a local barrier. Similarly to before, we construct an n-vertex ordered graph that does not contain a perfect H-tiling and whose minimum degree is at least $(1-1/\chi _{cr}^*(H))n-1$ , thereby giving the lower bound in cases (i) and (iii) of Theorem 1.8.

Extremal Example 2. Let H be an ordered graph and $n\in \mathbb {N}$ . Set $\ell :=\left \lfloor \frac {n}{\chi _{cr}^*(H)}+1\right \rfloor $ . Define $F_2(H,n)$ to be the unordered complete $\lceil n/\ell \rceil $ -partite graph on n vertices such that all classes have size $\ell $ except one class of size at most $\ell $ .

It is easy to check that the minimum degree of $F_2(H,n)$ is

(3) $$ \begin{align} \delta(F_2(H,n))=n-\ell=n-\left\lfloor\frac{n}{\chi_{cr}^*(H)}+1\right\rfloor\geq\left(1-\frac{1}{\chi_{cr}^*(H)}\right)n-1. \end{align} $$

Additionally, there exists a certain ordering of the vertices of $F_2(H,n)$ such that the resulting ordered graph does not contain a perfect H-tiling:

Lemma 2.3. Let H be an ordered graph and $n\in \mathbb {N}$ such that $|H|$ divides n. There exists an interval labelling $\phi $ of $F_2(H,n)$ such that the ordered graph $(F_2(H,n),\phi )$ does not contain a perfect H-tiling.

Proof. The critical chromatic number of $F_2(H,n)$ is

$$ \begin{align*} \chi_{cr}(F_2(H,n))=\frac{n}{\ell}<\frac{n}{n/\chi_{cr}^*(H)}=\chi_{cr}^*(H). \end{align*} $$

It follows that $F_2(H,n)$ is not a bottlegraph of H. Hence, by definition, there exists a permutation $\sigma $ of $[\lceil n/\ell \rceil ]$ and an interval labelling $\phi $ of $F_2(H,n)$ with respect to $\sigma $ such that $(F_2(H,n),\phi )$ does not contain a perfect H-tiling.

Lemma 2.3 and (3) immediately imply the following corollary.

Corollary 2.4. Let H be an ordered graph. Then given any $n \in \mathbb N$ divisible by $|H|$ ,

$$ \begin{align*} \delta_<(H,n)\geq \delta(F_2(H,n))+1 \geq\left(1-\frac{1}{\chi_{cr}^*(H)}\right)n. \end{align*} $$

Next we give a general lower bound for $\delta _{cov}(H,n)$ , which is asymptotically sharp if the ordered graph H does not have a local barrier, thereby giving the lower bound in Theorem 1.10(i).

Extremal Example 3. Let H be an ordered graph and $n\in \mathbb {N}$ . Let $F_3(H,n)$ be the complete $(\chi _<(H)-1)$ -partite ordered graph on n vertices with parts of size as equal as possible.

It is easy to check that the minimum degree of $F_3(H,n)$ is

$$ \begin{align*} \delta(F_3(H,n))= n-\left\lceil\frac{n}{\chi_<(H)-1}\right\rceil>\left(1-\frac{1}{\chi_<(H)-1}\right)n-1. \end{align*} $$

As $\chi _<(F_3(H,n))< \chi _<(H)$ , $F_3(H,n)$ does not contain a copy of H and thus does not contain an H-cover. We therefore obtain the following result.

Lemma 2.5. Let H be an ordered graph and $n \in \mathbb N$ . Then

$$ \begin{align*} \delta_{cov}(H,n)\geq \delta(F_3(H,n))+1>\left(1-\frac{1}{\chi_<(H)-1}\right)n. \end{align*} $$

For n divisible by $\chi _<(H)-1$ , $F_3(H,n)$ also shows that the minimum degree threshold that ensures an almost perfect H-tiling in an n-vertex ordered graph is more than $(1-1/(\chi _<(H)-1))n$ . Thus, combined with Theorem 1.6, this immediately implies that, for all ordered graphs H,

(4) $$ \begin{align} \chi_<(H)-1 < \chi^*_{cr}(H). \end{align} $$

Actually, we close the section by proving an even stronger lower bound on $\chi _{cr}^*(H)$ .

Proposition 2.6 (A lower bound for $\chi _{cr}^*(H)$ )

Let H be an ordered graph on h vertices and $r:=\chi _<(H)$ . Then

$$ \begin{align*} \chi_{cr}^*(H)\geq(r-1)+\frac{r-1}{h-1}. \end{align*} $$

Proof. Let B be an arbitrary bottlegraph of H. It suffices to show that $\chi _{cr}(B)\geq (r-1)+\frac {r-1}{h-1}$ . If $\chi _{cr}(B)\geq r$ , then we are done (since $h\geq r$ ), so for the rest of the proof, we assume that $\chi _{cr}(B)<r$ . In particular, as (4) implies that $\chi _{cr}(B)>r-1$ , this means B has exactly r parts. Let $B_1$ denote the part of B of the smallest size. Pick any interval labelling $\phi $ of B; then there exists some $t\in \mathbb {N}$ such that the ordered blow-up $(B(t),\phi )$ contains a perfect H-tiling $\mathcal {H}$ . Since B has exactly r parts, it follows that every copy of H in $(B(t),\phi )$ intersects all parts of B. Hence,

$$ \begin{align*} |B_1(t)|\geq|\mathcal{H}|=\frac{|B(t)|}{|H|}=t|B|/h\implies|B_1|\geq|B|/h, \end{align*} $$

and so

$$ \begin{align*} \chi_{cr}(B)=(r-1)\frac{|B|}{|B|-|B_1|}\geq(r-1)\frac{|B|}{|B|-|B|/h}=(r-1)+\frac{r-1}{h-1}.\\[-40pt] \end{align*} $$

In Section 3.3, we give a family of ordered graphs H for which the lower bound on $\chi _{cr}^*(H)$ in Proposition 2.6 is tight.

3 Motivating examples

3.1 An example for Theorem 1.8(i)

Recall that Extremal Example 2 yields the lower bound in cases (i) and (iii) of Theorem 1.8. The argument in Lemma 2.3 is rather straightforward. This is because of the definition of $\chi ^*_{cr}(H)$ ; if one takes a complete multipartite graph G with $\chi _{cr}(G)<\chi ^*_{cr}(H)$ , then by definition, there is a vertex labelling of G so that the resulting ordered graph does not contain a perfect H-tiling.

Therefore, if one provides an argument that justifies why a bottlegraph of H is optimal, this equivalently can be translated into an argument that explains why an ordered graph is an extremal example for cases (i) and (iii) of Theorem 1.8. In this way, one can view $\chi ^*_{cr}(H)$ as ‘encoding’ properties of the extremal example.

In Section 11, we will compute $\chi ^*_{cr}(H)$ for various classes of ordered graphs H. Often these arguments will be somewhat involved; thus, in these cases, the reason the extremal example for Theorem 1.8 does not contain a perfect H-tiling is also ‘involved’. That is, in general, the reason extremal examples do not contain perfect H-tilings is not as immediate as Lemma 2.3 might suggest. We illustrate this point through the following example.

Example 3.1 (An example for Theorem 1.8(i))

Let $\ell \geq 2$ , and let H be the complete $3$ -partite ordered graph with parts $H_1<H_2<H_3$ of size $\ell ,1,\ell $ , respectively.

For H as in Example 3.1, we have that $\chi _{cr}^*(H)=(4\ell ^2-1)/\ell ^2>3=\chi _<(H)$ . (In fact, in Proposition 11.5, we compute $\chi _{cr}^*(F)$ for all complete $3$ -partite ordered graphs F.) Thus, for such H, we are in case (i) of Theorem 1.8, so

$$ \begin{align*} \delta_<(H,n)=\left(1-\frac{\ell^2}{4\ell^2-1}+o(1)\right)n. \end{align*} $$

We now describe an extremal example for Theorem 1.8 for such H. Let $n\in \mathbb {N}$ such that $|H|$ divides n and $n\geq 20$ . Let G be the complete $4$ -partite ordered graph on n vertices with parts $G_1<G_2<G_3<G_4$ , where

$$ \begin{align*} |G_1|=|G_2|=|G_3|=\left\lfloor\frac{n\ell^2}{4\ell^2-1}\right\rfloor+1\quad\text{and}\quad|G_4|= n-3\left\lfloor\frac{n\ell^2}{4\ell^2-1}\right\rfloor-3. \end{align*} $$

Note that $G_4$ is the smallest part since $|G_i|\geq n/4$ for $i=1,2,3$ and $|G_4|\leq n/4$ . In particular,

$$ \begin{align*} \delta(G)=n-\left\lfloor\frac{n\ell^2}{4\ell^2-1}\right\rfloor-1\geq n-\frac{n\ell^2}{4\ell^2-1}-1=\left(1-\frac{\ell^2}{4\ell^2-1}\right)n-1.\\[-16pt] \end{align*} $$

Suppose for a contradiction that G contains a perfect H-tiling $\mathcal {H}$ . Let $\mathcal {A}\subseteq \mathcal {H}$ be the set of copies of H in $\mathcal {H}$ that have exactly $\ell $ vertices in $G_1$ and set $\mathcal {B}:=\mathcal {H}\setminus \mathcal {A}$ . This immediately implies

(5) $$ \begin{align} |G_1|\geq \ell|\mathcal{A}|. \\[-16pt]\nonumber\end{align} $$

Note that if $H'\in \mathcal {A}$ , then $H'$ has at most $\ell +1$ vertices in $G_1\cup G_2$ , while if $H'\in \mathcal {B}$ , then $H'$ has at most $\ell $ vertices in $G_1\cup G_2$ . It follows that

(6) $$ \begin{align} |G_1|+|G_2|\leq (\ell+1)|\mathcal{A}|+\ell|\mathcal{B}|.\\[-16pt]\nonumber \end{align} $$

Combining (5) and (6) yields the following:

$$ \begin{align*} |\mathcal{A}|+|\mathcal{B}|\stackrel{(6)}{\geq}&\frac{|G_1|+|G_2|-|\mathcal{A}|}{\ell}\stackrel{(5)}{\geq}\frac{\ell|G_1|+\ell|G_2|-|G_1|}{\ell^2} \\ =&\frac{2\ell-1}{\ell^2}\left(\left\lfloor\frac{n\ell^2}{4\ell^2-1}\right\rfloor+1\right)>\frac{2\ell-1}{\ell^2}\left(\frac{n\ell^2}{4\ell^2-1}\right)=\frac{n}{2\ell+1}.\\[-16pt] \end{align*} $$

The above is a contradiction since

$$ \begin{align*}| \mathcal{A}|+|\mathcal{B}|=|\mathcal{H}|=\frac{|G|}{|H|}=\frac{n}{2\ell+1}.\\[-16pt] \end{align*} $$

Hence, G does not contain a perfect H-tiling.

Note that G is a ‘space barrier’ construction as our argument tells us that $G_1 \cup G_2$ is ‘too big’ to ensure a perfect H-tiling in G; moreover, the reason $G_1 \cup G_2$ is ‘too big’, whilst not difficult, is not obvious at first sight (i.e., we needed to consider how two types of copies of H intersect $G_1\cup G_2$ ).

Space barrier constructions occur in many other settings, too (e.g., the Kühn–Osthus perfect tiling theorem [Reference Kühn and Osthus20]). However, all previous graph space barrier constructions we are aware of have a different flavour to the above space barrier G. Indeed, previously known examples fail to contain the desired substructure due to some very immediate property that means one vertex class is ‘too small’ or ‘too big’.

In Section 11, we compute $\chi ^*_{cr}(H)$ precisely for several classes of ordered graphs. In particular, we give other ordered graphs H that fall into the case (i) of Theorem 1.8, namely all complete $3$ -partite ordered graphs and all complete r-partite ordered graphs whose smallest part is the first or last part (see Propositions 11.3 and 11.5).

3.2 An example for Theorem 1.8(ii)

The next example provides a family of ordered graphs that fall into case (ii) of Theorem 1.8.

Example 3.2 (An example for Theorem 1.8(ii))

Let $r,k\geq 3$ , and let H be the ordered graph with vertex set $V(H)=[(r-1)k+2]$ and edge set $E(H)=\{(1,(r-1)k+2)\}\cup \{((s-1)k+2,sk+2):s\in [r-1]\}$ (see Figure 1). So $\chi _<(H)=r$ .

Figure 1 The ordered graph H as in Example 3.2 for $r=4$ and $k=3$

Let B be the complete r-partite graph with parts $B_1,\dots ,B_r$ , where $|B_i|=k$ for $i\in [r-1]$ and $|B_r|= 2$ . Observe that $\chi _{cr}(B)=(r-1)+2/k$ . It is straightforward to check that for any permutation $\sigma $ of $[r]$ and any interval labelling $\phi $ of B with respect to $\sigma $ , the ordered graph $(B,\phi )$ contains a spanning copy of H; hence B is a simple bottlegraph of H. It follows that

$$ \begin{align*} \chi_{cr}^*(H)\leq\chi_{cr}(B)=(r-1)+\frac{2}{k}<r=\chi_<(H).\\[-16pt] \end{align*} $$

Furthermore, H has a local barrier: for any interval $(r+1)$ -colouring $\{1\}<V_1<\cdots <V_r$ of H, we have that $(r-1)k+2\in V_r$ , and thus there is one edge between $\{1\}$ and $V_r$ .

3.3 An example for Theorem 1.8(iii)

Next we consider a family of ordered graphs that fall into case (iii) of Theorem 1.8.

Example 3.3 (An example for Theorem 1.8(iii))

Let $r,k\geq 2$ , and let H be the ordered graph with vertex set $V(H)=[(r-1)k+1]$ and edge set $E(H)=\{((s-1)k+1,sk+1):s\in [r-1]\}$ . So H is the path $(1)(k+1)(2k+1)\dots ((r-1)k+1)$ and $\chi _<(H)=r$ (see Figure 2).

Figure 2 The ordered graph H as in Example 3.3 for $r=4$ and $k=2$

We will explicitly compute $\chi ^*_{cr}(H)$ . In particular, we prove that $\chi ^*_{cr}(H)<\chi _<(H)$ and that H does not have a local barrier. We first construct a bottlegraph of H. Let B denote the complete r-partite graph with parts $B_1,\dots ,B_r$ , where $|B_i|=k$ for $i\in [r-1]$ and $|B_r|= 1$ . Observe that $\chi _{cr}(B)=(r-1)+1/k$ . It is straightforward to check that for any permutation $\sigma $ of $[r]$ and any interval labelling $\phi $ of B with respect to $\sigma $ , the ordered graph $(B,\phi )$ contains a spanning copy of H. Thus B is a simple bottlegraph of H, so $\chi _{cr}^*(H)\leq \chi _{cr}(B)=(r-1)+1/k$ . Moreover, Proposition 2.6 implies that in fact $\chi _{cr}^*(H)= (r-1)+1/k$ .

Finally, we show that H does not have a local barrier. Let $i\not =j\in [r+1]$ . If $i\not \in \{1,r+1\}$ , there exists an interval $(r+1)$ -colouring $V_1<\cdots <V_{r+1}$ of H such that $V_i=\{x\}$ with $x\not =(s-1)k+1$ for every $s\in [r]$ . Then x is isolated in H, so clearly there is no edge between x and $V_j$ . If $i=1$ , there exists an interval $(r+1)$ -colouring $V_1<\cdots <V_{r+1}$ of H such that $V_i=\{1\}$ and $V_j=\emptyset $ ; so again, there is no edge between $V_i$ and $V_j$ . The case $i=r+1$ is analogous; we take $V_i=\{(r-1)k+1\}$ and $V_j=\emptyset $ .

4 Proof of Theorem 1.8

In this section, we present some intermediate results and explain how they imply Theorem 1.8. Crucial to our approach will be the use of the absorbing method, a technique that was introduced systematically by Rödl, Ruciński and Szemerédi [Reference Rödl, Ruciński and Szemerédi25] but that has roots in earlier work (see, e.g., [Reference Krivelevich19]). Given ordered graphs $G,H$ and a set $W\subseteq V(G)$ , a set $S\subseteq V(G)$ is called an H-absorbing set for W if both $G[S]$ and $G[W\cup S]$ contain perfect H-tilings. In [Reference Balogh, Li and Treglown3, Theorem 4.1], Balogh, Li and the second author provided a minimum degree condition that ensures an ordered graph G contains a set $Abs$ that is an H-absorbing set for every not too large set $W\subseteq V(G)\setminus Abs$ .

Theorem 4.1 (Balogh, Li and Treglown [Reference Balogh, Li and Treglown3])

Let H be an ordered graph on h vertices, and let $\eta>0$ . Then there exists an $n_0\in \mathbb {N}$ and $0<\nu \ll \eta $ so that the following holds. Suppose that G is an ordered graph on $n\geq n_0$ vertices and

$$ \begin{align*} \delta(G)\geq\left(1-\frac{1}{\chi_<(H)}+\eta\right)n. \end{align*} $$

Then $V(G)=[n]$ contains a set $Abs$ so that

  • $|Abs|\leq \nu n$ ;

  • $Abs$ is an H-absorbing set for every $W\subseteq V (G)\setminus Abs$ such that $|W|\in h\mathbb {N}$ and $|W|\leq \nu ^3n$ .

Theorems 1.6 and 4.1 can be combined to yield a minimum degree condition that forces a perfect H-tiling. Indeed, let G and H be ordered graphs, and suppose that

$$ \begin{align*} \delta(G)\geq\left(1-\frac{1}{\max\{\chi_<(H),\chi_{cr}^*(H)\}}+o(1)\right)n. \end{align*} $$

We first invoke Theorem 4.1 to find a set $Abs\subseteq V(G)$ , which is an H-absorbing set for any not too large set $W\subseteq V(G)\setminus Abs$ . Then we apply Theorem 1.6 to $G\setminus Abs$ to find an H-tiling $\mathcal {M}_1$ , which covers all but a small proportion of vertices in $G\setminus Abs$ . Let W denote the set of such vertices in $G\setminus Abs$ . Since W is relatively small, $Abs$ is an H-absorbing set for W, and thus $G[W\cup Abs]$ contains a perfect H-tiling $\mathcal {M}_2$ . Finally, observe that $\mathcal {M}_1\cup \mathcal {M}_2$ is a perfect H-tiling in G.

Thus we have proven that

$$ \begin{align*} \delta_<(H,n)\leq\left(1-\frac{1}{\max\{\chi_<(H),\chi_{cr}^*(H)\}}+o(1)\right)n. \end{align*} $$

In particular, this is asymptotically sharp if $\chi _{cr}^*(H)\geq \chi _<(H)$ (by Corollary 2.4) or if $\chi _{cr}^*(H)<\chi _<(H)$ and H has a local barrier (by Corollary 2.2), therefore proving cases (i) and (ii) of Theorem 1.8. However, if $\chi _{cr}^*(H)<\chi _<(H)$ and H does not have a local barrier, then this minimum degree condition can be substantially lowered. To achieve this, we need a new absorbing result:

Theorem 4.2 (Absorbing theorem for nonlocal barriers)

Let H be an ordered graph on h vertices with $\chi _<(H)\geq 3$ , and let $\eta>0$ . If H does not have a local barrier and $\chi _{cr}^*(H)<\chi _<(H)$ , then there exists an $n_0\in \mathbb {N}$ and $0<\nu \ll \eta $ so that the following holds. Suppose that G is an ordered graph on $n\geq n_0$ vertices and

$$ \begin{align*} \delta(G)\geq\left(1-\frac{1}{\chi_<(H)-1}+\eta\right)n. \end{align*} $$

Then $V(G)=[n]$ contains a set $Abs$ so that

  • $|Abs|\leq \nu n$ ;

  • $Abs$ is an H-absorbing set for every $W\subseteq V(G)\setminus Abs$ such that $|W|\in h\mathbb {N}$ and $|W|\leq \nu ^3n$ .

Note that the statement of Theorem 4.2 is false if one allows $\chi _<(H)=2$ ; indeed, the conclusion of the theorem fails for so-called divisibility barriers H.Footnote 3 However, one can adapt our proof and relax the hypothesis of Theorem 4.2 to $\chi _<(H)\geq 2$ if one additionally assumes H is not a divisibility barrier. We will not do this in this paper, however, as [Reference Balogh, Li and Treglown3, Theorem 1.9] already resolves the perfect H-tiling problem for ordered graphs H with $\chi _<(H)=2$ .

We postpone the proof of Theorem 4.2 to Section 10. With Theorem 4.2 at hand, we can now give the proof of Theorem 1.8.

Proof of Theorem 1.8

First note that the lower bounds in parts (i)–(iii) of the theorem follow immediately from Corollary 2.4 (for (i) and (iii)) and Corollary 2.2 (for (ii)). Thus it remains to prove the upper bounds.

Let H be an ordered graph with $\chi _<(H)\geq 3$ , and let $\eta>0$ . Let $n\in \mathbb {N}$ be sufficiently large and such that $|H|$ divides n. Let G be an ordered graph on n vertices with minimum degree so that

  1. (i) $ \delta (G)\geq \left (1-\frac {1}{\chi ^*_{cr}(H)}+\eta \right )n $ if $\chi ^*_{cr}(H) \geq \chi _<(H)$ ;

  2. (ii) $ \delta (G)\geq \left (1-\frac {1}{\chi _<(H)}+\eta \right )n $ if $\chi ^*_{cr}(H) < \chi _<(H)$ and H has a local barrier;

  3. (iii) $\delta (G)\geq \left (1-\frac {1}{\chi ^*_{cr}(H)}+\eta \right )n $ if $\chi ^*_{cr}(H) < \chi _<(H)$ and H has no local barrier.

Recall that $\chi ^* _{cr}(H)> \chi _<(H)-1$ . Thus, by Theorem 4.1 (for cases (i) and (ii)) and Theorem 4.2 (for case (iii)), there exists some $0<\nu \ll \eta $ and a set $Abs\subseteq V(G)$ such that

  • $|Abs|\leq \nu n$ ;

  • $Abs$ is an H-absorbing set for every $W\subseteq V (G)\setminus Abs$ such that $|W|\in |H|\mathbb {N}$ and $|W|\leq \nu ^3n$ .

Let $G':=G\setminus Abs$ . In all cases, we have that $\delta (G')\geq (1-{1}/{\chi ^*_{cr}(H)} )|G'| .$

Since n was chosen to be sufficiently large, by Theorem 1.6, there exists an H-tiling $\mathcal {M}_1$ in $G'$ covering all but at most $\nu ^3n$ vertices. Let $W\subseteq V(G')$ denote the set of vertices not covered by $\mathcal {M}_1$ . Since $|H|$ divides n, $|V(\mathcal {M}_1)|$ and $|Abs|$ , we have that $|H|$ divides $|W|$ too. Also, $|W|\leq \nu ^3n$ , and hence $G'[W\cup Abs]$ contains a perfect H-tiling $\mathcal {M}_2$ . Finally, observe that $\mathcal {M}_1\cup \mathcal {M}_2$ is a perfect H-tiling of G, as desired.

5 Bottlegraphs

In the following proposition, we show that it suffices to consider bottlegraphs where all parts are of the same size except perhaps one smaller part.

Proposition 5.1. Let H be an ordered graph and B be a bottlegraph of H. There exist a bottlegraph $B'$ of H and an integer $m\in \mathbb {N}$ such that $\chi _{cr}(B')=\chi _{cr}(B)$ and all parts of $B'$ have size m except one part with size at most m.

Proof. Let B be a bottlegraph of H so B is a complete k-partite unordered graph with parts $B_1,\dots ,B_k$ for some $k\in \mathbb {N}$ . Without loss of generality, we may assume that $|B_1|\leq |B_i|$ for every $i>1$ . Let $B'$ be the complete k-partite unordered graph with parts $B^{\prime }_1,\dots ,B^{\prime }_k$ , where

$$ \begin{align*}|B^{\prime}_i|=\begin{cases} (k-1)|B_1| &\text{ if } i=1\\ |B|-|B_1| &\text{ otherwise.} \end{cases} \end{align*} $$

So $|B'|=(k-1)|B|$ . Furthermore, we have

$$ \begin{align*} |B^{\prime}_1|=(k-1)|B_1|=k|B_1|-|B_1|\leq|B|-|B_1|=|B^{\prime}_i| \end{align*} $$

for every $i>1$ . It follows that

$$ \begin{align*} \chi_{cr}(B')=\frac{(k-1)|B'|}{|B'|-|B^{\prime}_1|}=\frac{(k-1)^2|B|}{(k-1)|B|-(k-1)|B_1|}= \frac{(k-1)|B|}{|B|-|B_1|}=\chi_{cr}(B). \end{align*} $$

It remains to show that $B'$ is a bottlegraph of H. Observe that the vertices in $B^{\prime }_1$ can be partitioned into $(k-1)$ sets of size $|B_1|$ , while the vertices in $B^{\prime }_i$ can be partitioned into $(k-1)$ sets of sizes $|B_2|,\dots ,|B_k|$ , respectively, for every $i>1$ . This implies that $B'$ contains a perfect B-tiling consisting of $(k-1)$ copies of B.

Let $\{C_1,\dots ,C_{k-1}\}$ be a perfect B-tiling in $B'$ (i.e., each $C_i$ is a copy of B in $B'$ ). Let $\sigma $ be a permutation of $[k]$ , and let $\phi $ be an interval labelling of $B'$ with respect to $\sigma $ . For every $C_i$ , $\phi $ induces an interval labelling $\phi _i$ of $C_i$ with respect to some permutation $\sigma _i$ of $[k]$ . Since B is a bottlegraph, given any $i\in [k-1]$ , there exists some $t_i\in \mathbb {N}$ such that the ordered blow-up $(C_i(t_i),\phi _i)$ contains a perfect H-tiling. Set $t:=t_1t_2\dots t_{k-1}$ . Then the ordered blow-up $(C_i(t),\phi _i)$ contains a perfect H-tiling $\mathcal {M}_i$ , for each $i\in [k-1]$ .

Finally, $\mathcal {M}_1\cup \cdots \cup \mathcal {M}_{k-1}$ is a perfect H-tiling of the ordered blow-up $(B'(t),\phi )$ . Since $\sigma ,\phi $ were arbitrary, $B'$ is a bottlegraph of H.

Note that the notion of a bottlegraph of H (Definition 1.4) and $1$ -bottlegraph (Definition 1.14) are not quite the same. However, the next result implies that $\chi ^*_{cr}(H)=\chi ^*_{cr}(1,H)$ .

Proposition 5.2. Let H be an ordered graph. Then $\chi ^*_{cr}(H)=\chi ^*_{cr}(1,H)$ .

Proof. Let $\mathcal X:= \{ \chi _{cr}(B) : \ B \text { is a bottlegraph of } H\}$ and $\mathcal X_1:= \{ \chi _{cr}(B) : \ B \text { is a } 1\text {-bottlegraph of } H\}.$ Thus, $\inf \mathcal X= \chi ^*_{cr} (H)$ and $\inf \mathcal X_1= \chi ^*_{cr} (1,H)$ . By definition of a bottlegraph and $1$ -bottlegraph, we have that $\mathcal X_1 \subseteq \mathcal X$ ; so to prove the proposition, it suffices to show that $\mathcal X \subseteq \mathcal X_1$ .

Given any bottlegraph B of H, let $B'$ be the bottlegraph of H obtained by applying Proposition 5.1. So $B'$ satisfies conditions (i) and (ii) in the definition of a $1$ -bottlegraph of H and $\chi _{cr}(B')=\chi _{cr}(B)$ . As $B'$ is a bottlegraph of H, there is some $t\in \mathbb N$ so that $B'(t)$ satisfies condition (iii) of the definition of a $1$ -bottlegraph of H. Then $B'(t)$ is a $1$ -bottlegraph of H with $\chi _{cr}(B'(t))=\chi _{cr}(B')=\chi _{cr}(B)$ . Thus, $\mathcal X \subseteq \mathcal X_1$ , as desired.

6 The regularity lemma

In the proof of Theorem 4.2, we will make use of the regularity method. In this section, we state a multipartite version of Szemerédi’s regularity lemma and some other related tools. First we introduce some basic notation.

The density of an (ordered) bipartite graph with vertex classes A and B is defined to be

$$ \begin{align*} d(A,B):=\frac{e(A,B)}{|A||B|}. \end{align*} $$

Given $\varepsilon>0$ , a graph G and two disjoint sets $A, B\subset V(G)$ , we say that the pair $(A, B)_G$ (or simply $(A, B)$ when the underlying graph is clear) is $\varepsilon $ -regular if for all sets $X \subseteq A$ and $Y \subseteq B$ with $|X|\ge \varepsilon |A|$ and $|Y|\ge \varepsilon |B|$ , we have $|d(A,B)-d(X,Y)|< \varepsilon $ . Given $d\in [0, 1]$ , the pair $(A, B)_G$ is $(\varepsilon ,d)$ -regular if G is $\varepsilon $ -regular and $d(A,B)\geq d$ .

We now state some well-known properties of $\varepsilon $ -regular pairs. The first (see, e.g., [Reference Komlós and Simonovits18, Fact 1.5]) implies that one can delete many vertices from an $(\varepsilon ,d)$ -regular pair and still retain such a regularity property.

Lemma 6.1 (Slicing lemma)

Let $(A,B)_G$ be an $\varepsilon $ -regular pair of density d, and for some $\alpha>\varepsilon $ , let $A'\subseteq A$ , $B'\subseteq B$ with $|A'|\geq \alpha |A|$ and $|B'|\geq \alpha |B|$ . Then $(A', B')_G$ is $(\varepsilon ', d-\varepsilon )$ -regular with $\varepsilon ':=\max \{\varepsilon /\alpha , 2\varepsilon \}$ .

The following theorem is a multipartite version of Szemerédi’s regularity lemma [Reference Szemerédi27] (presented, e.g., as Lemma 5.5 in [Reference Balogh, Li and Treglown3]).

Theorem 6.2 (Multipartite regularity lemma)

Given any integer $t\geq 2$ , any $\varepsilon>0$ and any $\ell _0\in \mathbb {N}$ , there exists $L_0=L_0(\varepsilon , t, \ell _0)\in \mathbb {N}$ such that for every $d\in (0, 1]$ and for every nearly balanced t-partite graph $G=(W_1,\dots , W_t)$ on $n\geq L_0$ vertices, there exists an $\ell \in \mathbb {N}$ , a partition $W^0_i, W^1_i,\dots , W^{\ell } _i$ of $W_i$ for each $i\in [t]$ and a spanning subgraph $G'$ of G such that the following conditions hold:

  1. 1. $\ell _0\leq \ell \leq L_0$ ;

  2. 2. $d_{G'}(x)\geq d_G(x)-(d+\varepsilon )n$ for every $x\in V(G)$ ;

  3. 3. $|W^0_i|\leq \varepsilon n/t$ for every $i\in [t]$ ;

  4. 4. $|W^j_i| = |W^{j'}_{i'}|$ for every $i,i'\in [t]$ and $j,j'\in [\ell ]$ ;

  5. 5. For every $i,i'\in [t]$ and $j,j'\in [\ell ]$ either $(W^j_i, W^{j'}_{i'})_{G'}$ is an $(\varepsilon ,d)$ -regular pair or $G'[W^j_i, W^{j'}_{i'}]$ is empty.

We call the $W^j_i$ clusters, the $W^0_i$ the exceptional sets and the vertices in the $W^0_i$ exceptional vertices. We refer to $G'$ as the pure graph. The reduced graph R of G with parameters $\varepsilon $ , d and $\ell _0$ is the graph whose vertices are the $W^j_i$ (where $i \in [t]$ and $j\in [\ell ])$ and in which $W^j_i W^{j'}_{i'}$ is an edge precisely when $(W^j_i, W^{j'}_{i'})_{G'}$ is $(\varepsilon , d)$ -regular. The following well-known corollary of the regularity lemma shows that the reduced graph almost inherits the minimum degree of the original graph.

Proposition 6.3. Let $0<\varepsilon ,d,k<1$ , and let G be an n-vertex graph with $\delta (G)\geq kn$ . If R is the reduced graph of G obtained by applying Theorem 6.2 with parameters $\varepsilon ,d,\ell _0$ , then $\delta (R)\geq (k-2\varepsilon -d)|R|$ .

A useful tool to embed subgraphs into G using the reduced graph R is the so-called key lemma.

Lemma 6.4 (Key lemma [Reference Komlós and Simonovits18])

Let $0<\varepsilon <d$ and $q,t\in \mathbb {N}$ . Let R be a graph with $V(R)=\{v_1,\dots ,v_k\}$ . We construct a graph G as follows: replace every vertex $v_i\in V(R)$ with a set $V_i$ of q vertices, and replace each edge of R with an $(\varepsilon ,d)$ -regular pair. For each $v_i\in V(R)$ , let $U_i$ denote the set of t vertices in $R(t)$ corresponding to $v_i$ . Let H be a subgraph of $R(t)$ on h vertices with maximum degree $\Delta $ . Set $\delta :=d-\varepsilon $ and $\varepsilon _0:=\delta ^{\Delta }/(2+\Delta )$ . If $\varepsilon \leq \varepsilon _0$ and $t-1\leq \varepsilon _0q$ , then there are at least $(\varepsilon _0q)^h$ labelled copies of H in G so that if $x\in V(H)$ lies in $U_i$ in $R(t)$ , then x is embedded into $V_i$ in G.

As in [Reference Balogh, Li and Treglown3], some of our applications of Lemma 6.4 will take the following form: suppose that within an ordered graph G we have vertex classes $V_1<\ldots <V_k$ so that each pair $(V_i,V_j)_G$ is $(\varepsilon ,d)$ -regular. Then Lemma 6.4 tells us G contains many copies of any fixed size ordered graph H with $\chi _< (H)=k$ , where the ith vertex class of each such copy of H is embedded into $V_i$ .

7 Absorbing tools

In this section, we state some useful results for proving the existence of H-absorbing sets. The first result is the following crucial lemma of Lo and Markström [Reference Lo and Markström23]; we present the ordered version of their result, which appeared as Lemma 7.1 in [Reference Balogh, Li and Treglown3].

Lemma 7.1 (Lo and Markström [Reference Lo and Markström23])

Let $h,s\in \mathbb {N}$ and $\xi>0$ . Suppose that H is an ordered graph on h vertices. Then there exists an $n_0\in \mathbb {N}$ such that the following holds. Suppose that G is an ordered graph on $n\geq n_0$ vertices so that for any $x,y\in V(G)$ , there are at least $\xi n^{sh-1}$ $(sh-1)$ -sets $X\subseteq V(G)$ such that both $G[X\cup \{x\}]$ and $G[X\cup \{y\}]$ contain perfect H-tilings. Then $V(G)$ contains a set M so that

  • $|M|\leq (\xi /2)^hn/4$ ;

  • M is an H-absorbing set for any $W\subseteq V (G)\setminus M$ such that $|W|\in h\mathbb {N}$ and $|W|\leq (\xi /2)^{2h}n/(32s^2h^3)$ .

Informally, we will sometimes refer to a set X satisfying the assumptions of Lemma 7.1 as a chain of size $|X|$ between vertices x and y. The next lemma states that it is in some sense possible to concatenate chains.

Lemma 7.2. Let H be an (ordered) graph on h vertices, and let $\alpha ,\beta ,\gamma>0$ and $s_1,s_2, n\in \mathbb {N}$ , where

$$ \begin{align*} 0<1/n \ll \gamma\ll\alpha,\beta, 1/h, 1/s_1, 1/s_2. \end{align*} $$

Let G be an (ordered) graph on n vertices, $x,y\in V(G)$ and $A\subseteq V(G)\setminus \{x,y\}$ , where $|A|\geq \alpha n$ . Suppose that for every $z\in A$ , there exist at least $\beta n^{s_1h-1} \ (s_1h-1)$ -sets $X\subseteq V(G)$ such that both $G[X\cup \{x\}]$ and $G[X\cup \{z\}]$ contain perfect H-tilings and similarly there exist at least $\beta n^{s_2h-1} \ (s_2h-1)$ -sets $Y\subseteq V(G)$ such that both $G[Y\cup \{y\}]$ and $G[Y\cup \{z\}]$ contain perfect H-tilings. Then there exist at least $\gamma n^{(s_1+s_2)h-1} \ ((s_1+s_2)h-1)$ -sets $Z\subseteq V(G)$ such that both $G[Z\cup \{x\}]$ and $G[Z\cup \{y\}]$ contain perfect H-tilings.

Proof. Let $z\in A$ , and let $X,Y$ be an $(s_1h-1)$ -set and an $(s_2h-1)$ -set, respectively, which satisfy the above properties, with $X,Y$ disjoint so that $y \not \in X$ and $x \not \in Y$ . Then $X\cup \{z\}\cup Y$ is an $((s_1+s_2)h-1)$ -set and both $G[(X\cup \{z\}\cup Y)\cup \{x\}]$ and $G[(X\cup \{z\}\cup Y)\cup \{y\}]$ contain perfect H-tilings. Thus, it is enough to lower-bound the number of such triples $(z,X,Y)$ . There are at least $\alpha n$ choices for z. Given a fixed choice of z, there are at least $\beta n^{s_1h-1}-n^{s_1h-2}\geq \beta n^{s_1h-1}/2$ suitable choices for X. For a fixed X, there are at least $\beta n^{s_2h-1}-s_1hn^{s_2h-2}\geq \beta n^{s_2h-1}/2$ choices for Y. Therefore, in total, there are at least

$$ \begin{align*} \frac{(\alpha n)\cdot(\beta n^{s_1h-1}/2)\cdot(\beta n^{s_2h-1}/2)}