1. Introduction
For graphs $G$ and $H$ , we say that $G$ is Ramsey for $H$ , and write $G \rightarrow H$ , if every $2$ colouring of the edges of $G$ contains a monochromatic copy of $H$ . In 1978, Erdős, Faudree, Rousseau, and Schelp [Reference Erdős, Faudree, Rousseau and Schelp9] pioneered the study of the sizeRamsey number $\hat r(H)$ , defined as the smallest integer $m$ for which there exists a graph $G$ with $m$ edges such that $G \rightarrow H$ . The existence of the usual Ramsey number $r(H)$ shows that this notion is sensible, since, for any $H$ , it is easy to see that $\hat r(H) \leqslant \binom{r(H)}{2}$ . When $H$ is a complete graph, this inequality is an equality, a simple fact first observed by Chvátal.
An early example showing that sizeRamsey numbers can exhibit interesting behaviour was found by Beck [Reference Beck1], who showed that $P_n$ , the path with $n$ vertices, satisfies $\hat r(P_n) = O(n)$ , which is significantly smaller than the $O(n^2)$ bound that follows from applying the inequality above and the corresponding bound $r(P_n) = O(n)$ for the usual Ramsey number of $P_n$ . In a followup paper, Beck [Reference Beck2] asked whether a similar phenomenon occurs for all boundeddegree graphs, that is, whether, for any integer $\Delta \geqslant 3$ , there exists a constant $c$ such that any graph $H$ with $n$ vertices and maximum degree $\Delta$ has sizeRamsey number at most $cn$ . Although Rödl and Szemerédi [Reference Rödl and Szemerédi19] showed that this question has a negative answer already for $\Delta =3$ , much work has gone into extending Beck’s result to other natural families of graphs, including: cycles [Reference Haxell, Kohayakawa and Łuczak14], boundeddegree trees [Reference Friedman and Pippenger10], powers of paths and boundeddegree trees [Reference Berger, Kohayakawa and Maesaka3, Reference Clemens, Jenssen and Kohayakawa5, Reference Han, Jenssen, Kohayakawa, Mota and Roberts13], and more besides.
Most of the known families with linear sizeRamsey numbers have a bounded structural parameter, such as bandwidth [Reference Clemens, Jenssen and Kohayakawa5] or, more generally, treewidth [Reference Kamčev, Liebenau, Wood and Yepremyan15] (though see the recent papers [Reference Draganić, Krivelevich and Nenadov8, Reference Letzter, Pokrovskiy and Yepremyan18] for examples with a somewhat different flavour). However, a fairly simple family of graphs which does not fall into any of these categories, but may still have linear sizeRamsey numbers, is the family of twodimensional grid graphs. For $s \in{\mathbb{N}}$ , the $s \times s$ grid is the graph with vertex set $[s] \times [s]$ where two pairs are adjacent if and only if they differ by one in exactly one coordinate. Obviously, the maximum degree of the $s \times s$ grid is four, but its bandwidth and treewidth are both exactly $s$ (see, e.g., [Reference Chvátalová4]), so the problem of estimating the sizeRamsey number of this graph, and usually we will take $s = \sqrt{n}$ so that the graph has $n$ vertices, provides an interesting test case for exploring new ideas and techniques.
Regarding upper bounds for the sizeRamsey number of the $\sqrt{n} \times \sqrt{n}$ grid, an important result of Kohayakawa, Rödl, Schacht, and Szemerédi [Reference Kohayakawa, Rödl, Schacht and Szemerédi17], which says that every graph $H$ with $n$ vertices and maximum degree $\Delta$ satisfies $\hat r(H) \leqslant n^{21/\Delta +o(1)}$ , immediately yields the bound $n^{7/4+o(1)}$ . This was recently improved by Clemens, Miralaei, Reding, Schacht, and Taraz [Reference Clemens, Miralaei, Reding, Schacht and Taraz6] to $n^{3/2+o(1)}$ (and an alternative proof of this bound was also noted in our recent paper [Reference Conlon, Nenadov and Trujić7]). The goal of this short note is to provide an elementary proof of an improved upper bound.
Theorem 1.1 There exists a constant $C \gt 0$ such that the sizeRamsey number of the $\sqrt{n} \times \sqrt{n}$ grid graph is at most $Cn^{5/4}$ .
Like much of the work on sizeRamsey numbers, the previous bounds for grids were obtained by applying the sparse regularity method to show that every $2$ colouring of the edges of the Erdős–Rényi random graph $G_{n, p}$ , for some appropriate density $p$ , contains a monochromatic copy of the grid. However, it is a simple exercise in the first moment method to show that for $p \ll n^{1/2}$ the random graph $G_{n, p}$ with high probability does not contain the $s \times s$ grid graph as a subgraph if $s = \Theta (\sqrt{n})$ , so the bound $O(n^{3/2})$ is the best that one can hope to achieve using this procedure.
To see how it is that we gain on this bound, suppose that $s = \sqrt{n}$ . It is known [Reference Haxell, Kohayakawa and Łuczak14] that there are $K, \Delta \gt 0$ and a graph $H$ with $Ks$ vertices and maximum degree at most $\Delta$ which is Ramsey for $C_s$ , the cycle of length $s$ . Consider now a ‘blowup’ $\Gamma$ of $H$ obtained by replacing every $x \in V(H)$ by an independent set $V_x$ of order $\Theta (s)$ and every $xy \in H$ by a bipartite graph $(V_x,V_y)$ in which every edge exists independently with probability $p = \Theta (s^{1/2})$ . With high probability, such a blowup contains $\Theta (s^{5/2}) = \Theta (n^{5/4})$ edges. That is, instead of revealing a random graph $G_{n, p}$ on all $n = \Theta (s^2)$ vertices, we only reveal edges that lie within $\Theta (s)$ bipartite subgraphs, each with parts of order $\Theta (s)$ . This salvages a significant number of edges which would otherwise go to waste.
Consider now a $2$ colouring of $\Gamma$ and recall that $H$ was chosen so that $H \rightarrow C_s$ . A key lemma, Lemma 2.3 below, then allows us to conclude that there are sets $V_1,\dotsc,V_s$ in $\Gamma$ and a collection $U_i \subseteq V_i$ of large subsets such that all $(U_i,U_{i+1})$ with $i \in [s]$ , where addition is taken modulo $s$ , are ‘regular’ in the same colour. We may then sequentially embed the vertices of the grid so that the first row is embedded into $U_1,\dotsc,U_s$ , the second into $U_2,\dotsc,U_s,U_1$ , and so on.
2. Definitions and key lemmas
In this section, we recall several standard definitions and note two key lemmas that will be needed in the proof of Theorem 1.1. Most of these revolve around the concept of sparse regularity (for a thorough overview of which we refer the reader to the survey by Gerke and Steger [Reference Gerke and Steger12]).
For ${\varepsilon } \gt 0$ and $p \in (0, 1]$ , a pair of sets $(V_1, V_2)$ is said to be $({\varepsilon }, p)$ lowerregular in a graph $G$ if, for all $U_i \subseteq V_i$ , $i \in \{1,2\}$ , with $U_i \geqslant{\varepsilon }V_i$ , the density $d_G(U_1, U_2) = e_G(U_1, U_2)/(U_1U_2)$ of edges between $U_1$ and $U_2$ satisfies
Immediately from this definition, we get that in every $({\varepsilon },p)$ lowerregular pair $(V_1, V_2)$ , for each $i \in \{1,2\}$ , all but at most ${\varepsilon }V_i$ vertices in $V_i$ have degree at least $(1{\varepsilon })pV_{3i}$ into $V_{3i}$ – a fact we will make use of in the proof of Theorem 1.1. Another useful and wellknown property is that lowerregularity is inherited on large sets.
Lemma 2.1. Let $0 \lt{\varepsilon } \lt \delta$ , $p \in (0, 1]$ , and let $(V_1, V_2)$ be an $({\varepsilon }, p)$ lowerregular pair. Then any pair of subsets $V'_{\!\!i} \subseteq V_i$ , $i \in \{1,2\}$ , with $V'_{\!\!i} \geqslant \delta V_i$ form an $({\varepsilon }/\delta,p)$ lowerregular pair.
For $\lambda \gt 0$ and $p \in (0, 1]$ , a graph $G$ is said to be $(\lambda,p)$ uniform if, for all disjoint $X, Y \subseteq V(G)$ with $X, Y \geqslant \lambda V(G)$ , the density of edges between $X$ and $Y$ satisfies $d_G(X,Y) = (1\pm \lambda )p$ . If only the upper bound holds, the graph is said to be upperuniform.Footnote ^{1} For example, it is easy to see that the random graph $G_{n, p}$ is with high probability $(o(1),p)$ uniform whenever $p \gg 1/n$ . If $G = (V_1, V_2;\; E)$ is bipartite, we say that $G$ is $(\lambda, p)$ uniform or upperuniform if the same conditions hold for all $X \subseteq V_1$ and $Y \subseteq V_2$ with $X \geqslant \lambda V_1$ and $Y \geqslant \lambda V_2$ . In order to prove our main technical lemma, we rely on the following result, a simple corollary of [Reference Kohayakawa, Konstadinidis and Mota16, Lemma 6], whose proof follows a density increment argument. The same conclusion can also be obtained by an application of the sparse regularity lemma.
Lemma 2.2. For all $0 \lt{\varepsilon } \lt 1/2$ and $\alpha \in (0, 1)$ , there exists $\lambda \gt 0$ such that the following holds for every $p \in (0, 1]$ . Let $G = (V_1, V_2;\; E)$ be a $(\lambda, p)$ upperuniform bipartite graph with $V_1 = V_2$ and $E \geqslant \alpha V_1V_2p$ . Then there exist $U_i \subseteq V_i$ , $i \in \{1,2\}$ , with $U_i = \lambda V_i$ such that $(U_1, U_2)$ is $({\varepsilon }, \alpha p)$ lowerregular in $G$ .
The next lemma is the crux of our argument. Here and elsewhere, we say that $(X,Y;\; E)$ is lowerregular if $(X,Y)$ is lowerregular with respect to the set of edges $E$ .
Lemma 2.3. For every $r, \Delta \geqslant 2$ and ${\varepsilon } \gt 0$ , there exists $\lambda \gt 0$ such that the following holds for every $p \in (0, 1]$ . Let $H$ be a graph on at least two vertices with $\Delta (H) \leqslant \Delta$ and let $\Gamma$ be obtained by replacing every $x \in V(H)$ with an independent set $V_x$ of sufficiently large order $n$ and every $xy \in H$ by a $(\lambda,p)$ uniform bipartite graph between $V_x$ and $V_y$ . Then, for every $r$ colouring of the edges of $\Gamma$ , there exists an $r$ colouring $\varphi$ of the edges of $H$ and, for every $x \in V(H)$ , a subset $U_x \subseteq V_x$ of order $U_x = \lambda n$ such that $(U_x, U_y;\; E_{\varphi (xy)})$ is $({\varepsilon },p/(2r))$ lowerregular for each $xy \in H$ , where $E_{\varphi (xy)} \subseteq E(\Gamma )$ stands for the edges in colour $\varphi (xy)$ .
Proof. Given $\varepsilon$ , $r$ , and $\Delta$ , we let $\alpha = 1/(2r)$ , ${\varepsilon }_{\Delta +1} \;:\!=\;{\varepsilon }$ , $\lambda _{\Delta +1} = \lambda _{2.2}({\varepsilon }_{\Delta +1},\alpha )$ , and, for every $i = \Delta, \dotsc, 1$ , sequentially take ${\varepsilon }_i ={\varepsilon }_{i+1}\lambda _{i+1}$ and $\lambda _i = \lambda _{2.2}({\varepsilon }_i,\alpha )$ . Lastly, let $\lambda = \prod _{i \in [\Delta +1]} \lambda _i$ .
Fix any $r$ colouring of (the edges of) $\Gamma$ and, for every $c \in [r]$ , let $\Gamma _c$ stand for the subgraph (in terms of edges) in colour $c$ . Note that $H$ has edgechromatic number at most $\Delta +1$ . In other words, there exists a partition of the edges of $H$ into $H_1,\dotsc,H_{\Delta +1}$ such that each $H_i$ is a matching. We find the required collection $\{U_x\}_{x \in V(H)}$ by maintaining the following condition for every $i \in [\Delta +1]$ : for every $x \in V(H)$ , there exists a chain $V_x = U_x^0 \supseteq U_x^1 \supseteq U_x^2 \supseteq \dotsb \supseteq U_x^i$ such that

(i) $U_x^j = \lambda _jU_x^{j1}$ for all $j \in [i]$ and

(ii) for every $xy \in \bigcup _{j \leqslant i} H_j$ , $(U_x^i, U_y^i)$ is $({\varepsilon }_i,\alpha p)$ lowerregular in $\Gamma _c$ for some $c \in [r]$ .
Consequently, for $i = \Delta +1$ , we obtain sets $U_x \subseteq V_x$ , for every $x \in V(H)$ , of order $U_x = \left(\prod _{i \in [\Delta +1]} \lambda _i\right)\! n = \lambda n$ such that $(U_x,U_y)$ is $({\varepsilon }_{\Delta +1},\alpha p)$ lowerregular and, thus, $({\varepsilon },\alpha p)$ lowerregular for every $xy \in H$ . It remains to show that we can indeed do this.
Consider first $i = 1$ . For each $xy \in H_1$ , let $c \in [r]$ be the majority colour in $\Gamma [V_x,V_y]$ . As $e_{\Gamma _c}(V_x, V_y) \geqslant (1\lambda )n^2p/r$ , we may apply Lemma 2.2 with ${\varepsilon }_1$ (as $\varepsilon$ ) and $\Gamma _c[V_x,V_y]$ (as $G$ ) to obtain sets $U_x^1, U_y^1$ with the desired properties. For every $x \in V(H)$ which is isolated in $H_1$ , we simply take an arbitrary subset $U_x^1 \subseteq V_x$ of order $\lambda _1U_x^0$ . Thus, the required condition holds for $i=1$ .
Suppose now that the condition holds for some $i \geqslant 1$ and let us show that it also holds for $i+1$ . As above, for every $xy \in H_{i+1}$ , let $c \in [r]$ be the majority colour in $\Gamma [V_x,V_y]$ . Since $\Gamma [V_x, V_y]$ is $(\lambda,p)$ uniform and, by (i), $U_x^i, U_y^i \geqslant \lambda n$ , we have $e_\Gamma (U_x^i, U_y^i) = (1 \pm \lambda )U_x^iU_y^ip$ and, hence,
Lemma 2.2 applied to $\Gamma _c\!\left[U_x^i,U_y^i\right]$ with ${\varepsilon }_{i+1}$ (as $\varepsilon$ ) gives sets $U_x^{i+1} \subseteq U_x^i$ and $U_y^{i+1} \subseteq U_y^i$ of order
for which $\left(U_x^{i+1},U_y^{i+1}\right)$ is $({\varepsilon }_{i+1},\alpha p)$ lowerregular in $\Gamma _c$ . For every $x \in V(H)$ which is isolated in $H_{i+1}$ , we again take an arbitrary subset $U_x^{i+1} \subseteq U_x^i$ of order $\lambda _{i+1}U_x^i$ . Observe also that, for every $xz \in \bigcup _{j \leqslant i} H_j$ , since $(U_x^i,U_z^i)$ was $({\varepsilon }_i,\alpha p)$ lowerregular in $\Gamma_{c^{\prime}}$ for some $c' \in [r]$ and $U_x^{i+1} = \lambda _{i+1}U_x^i$ , Lemma 2.1 and the fact that ${\varepsilon }_i/\lambda _{i+1} ={\varepsilon }_{i+1}$ imply that $\left(U_x^{i+1},U_z^{i+1}\right)$ is $({\varepsilon }_{i+1},\alpha p)$ lowerregular in $\Gamma_{c^{\prime}}$ , as desired. This completes the proof.
We also need a variant of a result from our previous paper [Reference Conlon, Nenadov and Trujić7, Lemma 3.5] about regularity inheritance. While that result was stated for the usual (full) notion of regularity, we only need lowerregularity here, allowing us to save a factor of $(\log n)^{1/2}$ .
Lemma 2.4. For all ${\varepsilon }, \alpha, \lambda \gt 0$ , there exist positive constants ${\varepsilon }'({\varepsilon },\alpha )$ and $C({\varepsilon },\alpha,\lambda )$ such that for $p \geqslant Cn^{1/2}$ , with probability at least $1  o(n^{5})$ , the random graph $\Gamma \sim G_{n, p}$ has the following property.
Suppose $G \subseteq \Gamma$ and $V_1, V_2 \subseteq V(\Gamma )$ are disjoint subsets of order $\tilde n = \lambda n$ such that $(V_1, V_2)$ is $({\varepsilon }',\alpha p)$ lowerregular in $G$ . Then there exists $B \subseteq V(\Gamma )$ of order $B \leqslant{\varepsilon }\tilde n$ such that, for each $v, w \in V(\Gamma ) \setminus (V_1 \cup V_2 \cup B)$ (not necessarily distinct), the following holds: for any two subsets $N_v \subseteq N_\Gamma (v, V_1)$ and $N_w \subseteq N_\Gamma (w, V_2)$ of order $\alpha \tilde n p/4$ , both $(N_v, V_2)$ and $(N_v, N_w)$ are $({\varepsilon },\alpha p)$ lowerregular in $G$ .
Sketch of the proof. The proof proceeds along the same lines as the proof of [Reference Conlon, Nenadov and Trujić7, Lemma 3.5]. The only difference is that there we made use of an inheritance lemma for full regularity (namely, Corollary 3.5 in [Reference Škorić, Steger and Trujić20]), which requires the sets on which regularity is inherited to be of order at least $C\log n/p$ , resulting in the requirement that $p \geqslant C(\log n/n)^{1/2}$ . However, for lowerregularity, one can instead use the inheritance lemma of Gerke, Kohayakawa, Rödl, and Steger [Reference Gerke, Kohayakawa, Rödl and Steger11, Corollary 3.8], which only requires the sets to be of order at least $C/p$ , resulting in $p \geqslant Cn^{1/2}$ . The rest of the proof remains exactly the same.
3. Proof of Theorem 1.1
Since it requires no additional work, we will actually prove the $r$ colour analogue of Theorem 1.1. More precisely, we will show that for every integer $r \geqslant 2$ there exists a graph of order $n$ with $O(n^{5/4})$ edges for which every $r$ colouring of the edges contains a monochromatic copy of the $\delta \sqrt{n} \times \delta \sqrt{n}$ grid for some $\delta \gt 0$ .
By a result of Haxell, Kohayakawa, and Łuczak [Reference Haxell, Kohayakawa and Łuczak14, Theorem 10], there exist constants $K, \Delta \gt 0$ , both depending only on $r$ , such that, for every sufficiently large $s \in{\mathbb{N}}$ , there is a graph $H$ on $Ks$ vertices with maximum degree at most $\Delta$ which has the property that every $r$ colouring of its edges contains a monochromatic copy of $C_\ell$ , the cycle of length $\ell$ , for every $\log s \ll \ell \leqslant s$ . Let
We show that the sizeRamsey number of the $\delta s \times \delta s$ grid is $O(s^{5/2})$ , which, for $s = \sqrt{n}$ , implies the desired statement.
Let $\Gamma$ be a graph obtained by replacing every vertex $x \in V(H)$ by an independent set $V_x$ of order $s$ and every edge $xy \in H$ by a bipartite graph between $V_x$ and $V_y$ in which each edge exists independently with probability $p = Cs^{1/2}$ for some sufficiently large constant $C \gt 0$ . With high probability, $\Gamma$ has the following property:

(A1) $e_\Gamma (V'_{\!\!x}, V'_{\!\!y}) = (1\pm \lambda )V'_{\!\!x}V'_{\!\!y} p$ for every $xy \in H$ and $V'_{\!\!x} \subseteq V_x$ and $V'_{\!\!y} \subseteq V_y$ with $V'_{\!\!x}V'_{\!\!y}p \geqslant 100s/\lambda ^2$ .
This is a standard feature of random graphs and follows from the Chernoff bound together with an application of the union bound. In particular, it establishes that with high probability $\Gamma [V_x,V_y]$ is $(\lambda,p)$ uniform for every $xy \in H$ and, therefore, $\Gamma$ has at most
edges. Additionally, with high probability, $\Gamma$ is such that every $\Gamma [V_x \cup V_y \cup V_z]$ has the property of Lemma 2.4 (applied with ${\varepsilon }/9$ as $\varepsilon$ , $\lambda/3$ as $\lambda$ , and $3s$ as $n$ ) for every path $xyz$ of length two in $H$ .Footnote ^{2} This again follows from the union bound, as there are $O(s)$ such paths in total and the conclusion of Lemma 2.4 holds with probability $1  o(s^{5})$ for every fixed path. We now fix an outcome of $\Gamma$ which satisfies all of these properties.
Consider some $r$ colouring of the edges of $\Gamma$ and let $\varphi$ be the colouring of the edges of $H$ given by Lemma 2.3 (applied with $\varepsilon '$ as $\varepsilon$ ). By the choice of $H$ , this colouring contains a monochromatic copy of $C_{\delta s}$ , which, without loss of generality, we may assume has vertices $1,\dotsc,\delta s$ . Therefore, there is a colour $c \in [r]$ and sets $U_i$ of order $\tilde s = \lambda s$ in $\Gamma$ such that, for every $i \in [\delta s]$ , the pair $(U_i,U_{i+1})$ is $({\varepsilon }',\alpha p)$ lowerregular in the subgraph of $\Gamma$ induced by colour $c$ , where we identify $\delta s + i$ with $i$ . Let $G$ be the graph induced by these sets whose edges are the edges of $\Gamma$ of colour $c$ . We will show that $G$ contains the $\delta s \times \delta s$ grid as a subgraph.
For every $i \in [\delta s]$ , let $B \subseteq U_i \cup U_{i+1} \cup U_{i+2}$ be the set given by Lemma 2.4 (which was applied with ${\varepsilon }/9$ as $\varepsilon$ , $\lambda/3$ as $\lambda$ , and $3s$ as $n$ ) on $\Gamma [U_i \cup U_{i+1} \cup U_{i+2}]$ , which is a set of ‘bad vertices’ for the pair $(U_{i+1}, U_{i+2})$ . As each $U_i$ is a part of three such applications, by the chosen properties of $\Gamma$ , for every $i \in [\delta s]$ there exists a set $B_i \subseteq U_i$ of order $B_i \leqslant{\varepsilon }\tilde s$ such that:

(B1) $(N_v, U_{i+2} \setminus B_{i+2})$ is $({\varepsilon }, \alpha p)$ lowerregularFootnote ^{3} in $G$ for every $v \in U_i \setminus B_i$ and $N_v \subseteq N_G(v, U_{i+1})$ of order $\alpha \tilde sp/4$ and

(B2) $(N_v, N_u)$ is $({\varepsilon }, \alpha p)$ lowerregular in $G$ for every $v \in U_i \setminus B_i$ , $u \in U_{i+1} \setminus B_{i+1}$ and $N_v \subseteq N_G(v, U_{i+1})$ , $N_u \subseteq N_G(u, U_{i+2})$ , each of order $\alpha \tilde sp/4$ .
Our plan is to embed the vertex $(i, j)$ of the $\delta s \times \delta s$ grid into $U_{i+j1}$ . The next claim helps us achieve this.
Claim 3.1 Let $i \in [\delta s]$ . Suppose that sets $S_{i+j1} \subseteq U_{i+j1} \setminus B_{i+j1}$ of order $\alpha \tilde sp/4$ are given for each $j \in [\delta s]$ and that $(S_{i+j1}, S_{i+j})$ and $(S_{i+j1}, U_{i+j} \setminus B_{i+j})$ are $({\varepsilon }, \alpha p)$ lowerregular. Then, for every $Q_{i+j1} \subseteq U_{i+j1}$ , $j \in [\delta s]$ , of order $Q_{i+j1} \leqslant 2{\varepsilon } \tilde s$ , there exists a path $v_1,\dotsc,v_{\delta s}$ with each $v_j \in S_{i+j1}$ such that $N_G(v_j, U_{i+j} \setminus Q_{i+j}) \geqslant \alpha \tilde sp/4$ .
Before proving the claim, we show how to complete the embedding of the grid assuming that it holds. We start by embedding the first row. Let $v_1 \in U_1 \setminus B_1$ be a vertex for which there is $S_2 \subseteq N_G(v_1, U_2 \setminus B_2)$ of order $\alpha \tilde sp/4$ such that $(S_2, U_3 \setminus B_3)$ is $({\varepsilon }, \alpha p)$ lowerregular. As $(U_1 \setminus B_1, U_2 \setminus B_2)$ is $(2{\varepsilon }', \alpha p)$ lowerregular, there are at least $(12{\varepsilon }')(1{\varepsilon })\tilde s$ vertices $v \in U_1 \setminus B_1$ that satisfy
by our choice of constants. Thus, by property () almost any choice of $v_1 \in U_1 \setminus B_1$ will do. Sequentially, for every $i \geqslant 2$ , let $v_i \in S_i$ be a vertex for which there is $S_{i+1} \subseteq N_G(v_i, U_{i+1} \setminus B_{i+1})$ of order $\alpha \tilde sp/4$ and both $(S_{i+1}, U_{i+2} \setminus B_{i+2})$ and $(S_i, S_{i+1})$ are $({\varepsilon }, \alpha p)$ lowerregular. This is possible as $(S_i, U_{i+1} \setminus B_{i+1})$ is $({\varepsilon }, \alpha p)$ lowerregular and properties () and () hold. We continue until we have embedded the first row of the grid as $v_1,\dotsc,v_{\delta s}$ , with $v_i \in U_i$ for every $i \in [\delta s]$ .
Consider now sets $S_2, \dotsc, S_{\delta s}, S_1$ which we previously chose, where we note that $S_1$ was defined when we embedded $v_{\delta s}$ . In particular, $S_{1+j} \subseteq U_{1+j} \setminus B_{1+j}$ and $(S_{1+j}, S_{2+j})$ and $(S_{1+j}, U_{2+j} \setminus B_{2+j})$ are both $({\varepsilon }, \alpha p)$ regular for every $j \in [\delta s]$ . Then, by setting $Q_{1+j} \;:\!=\; B_{1+j} \cup \{v_{1+j}\}$ and invoking Claim 3.1 with $i = 2$ , we can embed the second row of the grid as $u_1, \dotsc, u_{\delta s}$ , with $u_j \in S_{1+j}$ for every $j \in [\delta s]$ . By the conclusion of Claim 3.1 and a slight abuse of notation, there is a collection of sets $S_{2+j} \subseteq N_G(u_j, U_{2+j} \setminus Q_{2+j})$ for every $j \in [\delta s]$ , each of order $\alpha \tilde sp/4$ , which, by () and (), as $u_j \in U_{1+j} \setminus B_{1+j}$ and $u_{j+1} \in U_{1+j+1} \setminus B_{1+j+1}$ , are such that $(S_{2+j}, S_{2+j+1})$ and $(S_{2+j}, U_{2+j+1} \setminus B_{2+j+1})$ are $({\varepsilon }, \alpha p)$ lowerregular.
The same process can now be repeated for any $i \geqslant 3$ (see Figure 1) by setting the sets $Q_{i+j1} \subseteq U_{i+j1}$ for every $j \in [\delta s]$ to be the union of $B_{i+j1}$ and the vertices of the grid that were previously embedded into $U_{i+j1}$ , that is, the images of the vertices $(1,i+j1), (2,i+j2), \dotsc, (i1,j+1)$ . Since $B_{i+j1} \leqslant{\varepsilon }\tilde s$ , $\delta \lt{\varepsilon }\lambda$ , and the lowerregularity conditions hold by () and (), we may apply Claim 3.1 to embed the $i$ th row. It only remains to prove this claim.
Proof of Claim 3.1. Without loss of generality, we may assume that all the $Q_{i+j1}$ are of order $2{\varepsilon }\tilde s$ , as we can take arbitrary supersets if this is not the case. Let $S'_{\!\!i+j1} \subseteq S_{i+j1}$ be the set of all $v \in S_{i+j1}$ with at least $\alpha \tilde sp/4$ neighbours in $U_{i+j} \setminus Q_{i+j}$ . On the one hand, as $(S_{i+j1}, U_{i+j} \setminus B_{i+j})$ is $({\varepsilon },\alpha p)$ lowerregular and, thus, there are fewer than ${\varepsilon }S_{i+j1}$ vertices in $S_{i+j1}$ with degree less than $\alpha \tilde sp/2$ in $U_{i+j} \setminus B_{i+j}$ , we have
On the other hand, assuming $S_{i+j1} \setminus S'_{\!\!i+j1}$ is of order at least $\alpha \tilde sp/16$ and, hence,
for $C \gt 0$ sufficiently large, property () implies that
Since ${\varepsilon } \lt \alpha/128$ , this is a contradiction. Therefore, there are sets $S'_{\!\!i+j1} \subseteq S_{i+j1}$ of order at least $S_{i+j1}  \alpha \tilde sp/16$ for each $j \in [\delta s]$ such that every $v \in S'_{\!\!i+j1}$ satisfies $N_G(v, U_{i+j} \setminus Q_{i+j}) \geqslant \alpha \tilde sp/4$ .
We will now find a collection of sets $S''_{\!\!\!i+j1} \subseteq S'_{\!\!i+j1}$ of order at least $S_{i+j1}  \alpha \tilde sp/8$ such that, for every $2 \leqslant j \leqslant \delta s$ , every $v \in S''_{\!\!\!i+j2}$ has a nonempty $N_G(v,S''_{\!\!\!i+j1})$ . First, choose $S''_{\!\!\!i+\delta s1} \subseteq S'_{\!\!i+\delta s1}$ of order $S_{i+\delta s1}  \alpha \tilde sp/8$ arbitrarily, noting that such a set exists by the bound on $S'_{\!\!i+\delta s1}$ . Having chosen $S''_{\!\!\!i+j1}$ for some $2 \leqslant j \leqslant \delta s$ , we choose $S''_{\!\!\!i+j2}$ as follows. Recall that $(S_{i+j2},S_{i+j1})$ is $({\varepsilon },\alpha p)$ lowerregular and, thus, by Lemma 2.1 and the bounds on the orders of $S'_{\!\!i+j2}$ and $S''_{\!\!\!i+j1}$ , $(S'_{\!\!i+j2},S''_{\!\!\!i+j1})$ is $(2{\varepsilon },\alpha p)$ lowerregular. It follows that there are at least $(12{\varepsilon })S'_{\!\!i+j2} \geqslant S_{i+j2}  \alpha \tilde sp/8$ vertices $v \in S'_{\!\!i+j2}$ which satisfy
We declare the set of such vertices to be $S''_{\!\!\!i+j2}$ and continue on to the next index $j$ .
Starting with an arbitrary $v_1 \in S''_{\!\!\!i}$ and sequentially choosing $v_j \in N_G(v_{j1}, S''_{\!\!\!i+j1})$ now completes the proof.