A family of sets is intersecting if no two of its members are disjoint, and has the Erdős–Ko–Rado property (or is EKR) if each of its largest intersecting subfamilies has non-empty intersection.

Denote by ${{\cal H}_k}(n,p)$ the random family in which each k-subset of {1, …, n} is present with probability p, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks:

\begin{equation} {\rm{For what }}p = p(n,k){\rm{is}}{{\cal H}_k}(n,p){\rm{likely to be EKR}}? \end{equation}

Here, for fixed c < 1/4, and $k \lt \sqrt {cn\log n} $ we give a precise answer to this question, characterizing those sequences p = p(n, k) for which

\begin{equation} {\mathbb{P}}({{\cal H}_k}(n,p){\rm{is EKR}}{\kern 1pt} ) \to 1{\rm{as }}n \to \infty . \end{equation}

We analyse the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H.

In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained.

For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a superpolynomial upper bound.

Let $F$ be an algebraically closed field of characteristic $0$ and let $\operatorname{sp}(2l,F)$ be the rank $l$ symplectic algebra of all $2l\times 2l$ matrices $x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$ over $F$ , where $A^{t}$ is the transpose of $A$ and $B,C$ are symmetric matrices of order $l$ . The commuting graph $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ of $\operatorname{sp}(2l,F)$ is a graph whose vertex set consists of all nonzero elements in $\operatorname{sp}(2l,F)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$ . We prove that the diameter of $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ is $4$ when $l>2$ .

In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree is unimodal. Although it attracts many researchers' attention, it is still open. Motivated by this conjecture, in this paper, we prove that rooted products of some graphs preserve real rootedness of independence polynomials. As application, we not only give a unified proof for some known results, but also we can apply them to generate infinite kinds of trees whose independence polynomials have only real zeros. Thus their independence polynomials are unimodal.

We provide a deterministic algorithm that finds, in ɛ-O(1)n2 time, an ɛ-regular Frieze–Kannan partition of a graph on n vertices. The algorithm outputs an approximation of a given graph as a weighted sum of ɛ-O(1) many complete bipartite graphs.

As a corollary, we give a deterministic algorithm for estimating the number of copies of H in an n-vertex graph G up to an additive error of at most ɛnv(H), in time ɛ-OH(1)n2.

An (improper) graph colouring has defect d if each monochromatic subgraph has maximum degree at most d, and has clustering c if each monochromatic component has at most c vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than (2d+2)/(d+2)k is k-choosable with defect d. This improves upon a similar result by Havet and Sereni (J. Graph Theory, 2006). For clustered choosability of graphs with maximum average degree m, no (1-ɛ)m bound on the number of colours was previously known. The above result with d=1 solves this problem. It implies that every graph with maximum average degree m is $\lfloor{\frac{3}{4}m+1}\rfloor$ -choosable with clustering 2. This extends a result of Kopreski and Yu (Discrete Math., 2017) to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degree m is $\lfloor{\frac{7}{10}m+1}\rfloor$ -choosable with clustering 9, and is $\lfloor{\frac{2}{3}m+1}\rfloor$ -choosable with clustering O(m). As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth–moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.

For $G$ a finite non-Abelian group we write $c(G)$ for the probability that two randomly chosen elements commute and $k(G)$ for the largest integer such that any $k(G)$ -colouring of $G$ is guaranteed to contain a monochromatic quadruple $(x,y,xy,yx)$ with $xy\neq yx$ . We show that $c(G)\rightarrow 0$ if and only if $k(G)\rightarrow \infty$ .

We study forcing pairs for quasirandom graphs. Chung, Graham, and Wilson initiated the study of families  ${\mathcal{F}}$ of graphs with the property that if a large graph  $G$ has approximately homomorphism density $p^{e(F)}$ for some fixed $p\in (0,1]$ for every $F\in {\mathcal{F}}$ , then  $G$ is quasirandom with density  $p$ . Such families ${\mathcal{F}}$ are said to be forcing. Several forcing families were found over the last three decades and characterizing all bipartite graphs  $F$ such that  $(K_{2},F)$ is a forcing pair is a well-known open problem in the area of quasirandom graphs, which is closely related to Sidorenko’s conjecture. In fact, most of the known forcing families involve bipartite graphs only.

We consider forcing pairs containing the triangle  $K_{3}$ . In particular, we show that if  $(K_{2},F)$ is a forcing pair, then so is $(K_{3},F^{\rhd })$ , where $F^{\rhd }$ is obtained from  $F$ by replacing every edge of $F$ by a triangle (each of which introduces a new vertex). For the proof we first show that  $(K_{3},C_{4}^{\rhd })$ is a forcing pair, which strengthens related results of Simonovits and Sós and of Conlon et al.

As usual, $P_{n}$ ( $n\geq 1$ ) denotes the path on $n$ vertices. The gem is the graph consisting of a $P_{4}$ together with an additional vertex adjacent to each vertex of the $P_{4}$ . A graph is called ( $P_{5}$ , gem)-free if it has no induced subgraph isomorphic to a $P_{5}$ or to a gem. For a graph $G$ , $\unicode[STIX]{x1D712}(G)$ denotes its chromatic number and $\unicode[STIX]{x1D714}(G)$ denotes the maximum size of a clique in $G$ . We show that $\unicode[STIX]{x1D712}(G)\leq \lfloor \frac{3}{2}\unicode[STIX]{x1D714}(G)\rfloor$ for every ( $P_{5}$ , gem)-free graph $G$ .

The strong chromatic number χs(G) of a graph G on n vertices is the least number r with the following property: after adding $r\lceil n/r\rceil-n$ isolated vertices to G and taking the union with any collection of spanning disjoint copies of Kr in the same vertex set, the resulting graph has a proper vertex colouring with r colours. We show that for every c > 0 and every graph G on n vertices with Δ(G) ≥ cn, χs(G) ≤ (2+o(1))Δ(G), which is asymptotically best possible.

We prove a lower bound on the difference between the spectral radius of the Cayley graph of a group $G$ and the spectral radius of the Schreier graph $H\backslash G$ for any subgroup $H$ . As an application, we extend Kesten’s theorem on spectral radii to uniformly recurrent subgroups and give a short proof that the result of Lyons and Peres on cycle density in Ramanujan graphs [Lyons and Peres. Cycle density in infinite Ramanujan graphs. Ann. Probab.43(6) (2015), 3337–3358, Theorem 1.2] holds on average. More precisely, we show that if ${\mathcal{G}}$ is an infinite deterministic Ramanujan graph then the time spent in short cycles by a random trajectory of length $n$ is $o(n)$ .

We prove the following 30 year-old conjecture of Győri and Tuza: the edges of every n-vertex graph G can be decomposed into complete graphs C1,. . .,Cℓ of orders two and three such that |C1|+···+|Cℓ| ≤ (1/2+o(1))n2. This result implies the asymptotic version of the old result of Erdős, Goodman and Pósa that asserts the existence of such a decomposition with ℓ ≤ n2/4.

Let $G$ be a finite group and let $\text{Irr}(G)$ be the set of all irreducible complex characters of $G$ . Let $\unicode[STIX]{x1D70C}(G)$ be the set of all prime divisors of character degrees of $G$ . The character degree graph $\unicode[STIX]{x1D6E5}(G)$ associated to $G$ is a graph whose vertex set is $\unicode[STIX]{x1D70C}(G)$ , and there is an edge between two distinct primes $p$ and $q$ if and only if $pq$ divides $\unicode[STIX]{x1D712}(1)$ for some $\unicode[STIX]{x1D712}\in \text{Irr}(G)$ . We prove that $\unicode[STIX]{x1D6E5}(G)$ is $k$ -regular for some natural number $k$ if and only if $\overline{\unicode[STIX]{x1D6E5}}(G)$ is a regular bipartite graph.

A tree functional is called additive if it satisfies a recursion of the form $F(T) = \sum_{j=1}^k F(B_j) + f(T)$ , where B1, …, Bk are the branches of the tree T and f (T) is a toll function. We prove a general central limit theorem for additive functionals of d-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalized plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group of d-ary increasing trees, but other examples (old and new) are covered as well.

For an edge-coloured graph G, the minimum colour degree of G means the minimum number of colours on edges which are incident to each vertex of G. We prove that if G is an edge-coloured graph with minimum colour degree at least 5, then V(G) can be partitioned into two parts such that each part induces a subgraph with minimum colour degree at least 2. We show this theorem by proving amuch stronger form. Moreover, we point out an important relationship between our theorem and Bermond and Thomassen’s conjecture in digraphs.

We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. One attempt to drop the bipartiteness assumption precipitates a natural class of Ramsey-type questions, of independent interest. For example, does every triangle-free graph of minimum degree d contain a bipartite induced subgraph of minimum degree Ω(log d) as d→∞?

An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a subset of {1,…, n} is a (βn, βn, βn) -array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants α, β > 0 such that, for every positive integer n, if P is an α-dense n × n partial Latin square, A is an n × n (βn, βn, βn)-array, and no cell of P contains a symbol that appears in the corresponding cell of A, then there is a completion of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty cell of P, and, for each i, j satisfying 1 ≤ i, j ≤ n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.

We investigate bounds in Ramsey’s theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek (Duke Mathematical Journal163(12) (2014), 2243–2270) from the semialgebraic case to arbitrary polynomially bounded $o$ -minimal expansions of $\mathbb{R}$ , and show that it does not hold in $\mathbb{R}_{\exp }$ . This provides a new combinatorial characterization of polynomial boundedness for $o$ -minimal structures. We also prove an analog for relations definable in $P$ -minimal structures, in particular for the field of the $p$ -adics. Generalizing Conlon et al. (Transactions of the American Mathematical Society366(9) (2014), 5043–5065), we show that in distal structures the upper bound for $k$ -ary definable relations is given by the exponential tower of height $k-1$ .

Let $n,r,k\in \mathbb{N}$ . An $r$ -colouring of the vertices of a regular $n$ -gon is any mapping $\unicode[STIX]{x1D712}:\mathbb{Z}_{n}\rightarrow \{1,2,\ldots ,r\}$ . Two colourings are equivalent if one of them can be obtained from another by a rotation of the polygon. An $r$ -ary necklace of length $n$ is an equivalence class of $r$ -colourings of $\mathbb{Z}_{n}$ . We say that a colouring is $k$ -alternating if all $k$ consecutive vertices have pairwise distinct colours. We compute the smallest number $r$ for which there exists a $k$ -alternating $r$ -colouring of $\mathbb{Z}_{n}$ and we count, for any $r$ , 2-alternating $r$ -colourings of $\mathbb{Z}_{n}$ and 2-alternating $r$ -ary necklaces of length $n$ .

An embedding of a metric graph $(G,d)$ on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_{e}(G)$ of $(G,d)$ is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute $g_{e}(G)$ and show that, for every integer $g\geq g_{e}(G)$ , there is an embedding of $(G,d)$ (possibly after a rescaling of $d$ ) on a surface of genus $g$ . Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$ . The maximum essential genus $g_{e}^{\max }(G)$ of $(G,d)$ is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of $(G,d)$ , where $g_{e}(G)$ and $g_{e}^{\max }(G)$ are realised.