Abstract The well-known Erdős-Hajnal conjecture states that for any graph $F$ , there exists $\epsilon \gt 0$ such that every $n$ -vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^{\epsilon }$ . We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$ , then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$ , for every $S \subseteq \{0,1,2,3,4\}$ , we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in $S$ . In most cases the bounds are essentially tight. We also determine, for all $S$ , whether the growth rate is polynomial or polylogarithmic. Some open problems remain.

Abstract The factorially normalized Bernoulli polynomials$b_n(x) = B_n(x)/n!$are known to be characterized by$b_0(x) = 1$and$b_n(x)$for$n \gt 0$is the anti-derivative of$b_{n-1}(x)$subject to$\int _0^1 b_n(x) dx = 0$. We offer a related characterization:$b_1(x) = x - 1/2$and$({-}1)^{n-1} b_n(x)$for$n \gt 0$is the$n$-fold circular convolution of$b_1(x)$with itself. Equivalently,$1 - 2^n b_n(x)$is the probability density at$x \in (0,1)$of the fractional part of a sum of$n$independent random variables, each with the beta$(1,2)$probability density$2(1-x)$at$x \in (0,1)$. This result has a novel combinatorial analog, theBernoulli clock: mark the hours of a$2 n$hour clock by a uniformly random permutation of the multiset$\{1,1, 2,2, \ldots, n,n\}$, meaning pick two different hours uniformly at random from the$2 n$hours and mark them$1$, then pick two different hours uniformly at random from the remaining$2 n - 2$hours and mark them$2$, and so on. Starting from hour$0 = 2n$, move clockwise to the first hour marked$1$, continue clockwise to the first hour marked$2$, and so on, continuing clockwise around the Bernoulli clock until the first of the two hours marked$n$is encountered, at a random hour$I_n$between$1$and$2n$. We show that for each positive integer$n$, the event$( I_n = 1)$has probability$(1 - 2^n b_n(0))/(2n)$, where$n! b_n(0) = B_n(0)$is the$n$th Bernoulli number. For$ 1 \le k \le 2 n$, the difference$\delta _n(k)\,:\!=\, 1/(2n) -{\mathbb{P}}( I_n = k)$is a polynomial function of$k$with the surprising symmetry$\delta _n( 2 n + 1 - k) = ({-}1)^n \delta _n(k)$, which is a combinatorial analog of the well-known symmetry of Bernoulli polynomials$b_n(1-x) = ({-}1)^n b_n(x)$.

Abstract A graph is called $k$ -critical if its chromatic number is $k$ but every proper subgraph has chromatic number less than $k$ . An old and important problem in graph theory asks to determine the maximum number of edges in an $n$ -vertex $k$ -critical graph. This is widely open for every integer $k\geq 4$ . Using a structural characterisation of Greenwell and Lovász and an extremal result of Simonovits, Stiebitz proved in 1987 that for $k\geq 4$ and sufficiently large $n$ , this maximum number is less than the number of edges in the $n$ -vertex balanced complete $(k-2)$ -partite graph. In this paper, we obtain the first improvement in the above result in the past 35 years. Our proofs combine arguments from extremal graph theory as well as some structural analysis. A key lemma we use indicates a partial structure in dense $k$ -critical graphs, which may be of independent interest.

Abstract We extend a recent argument of Kahn, Narayanan and Park ((2021) Proceedings of the AMS 149 3201–3208) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of $C_4$ that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold for edge-overlapping spanning $K_r$ -cycles.

Abstract We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most $k - 1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$ . The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and Füredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $k\geq 2^{n-d-1}$ we have $f(n,k,d)=2^d k-\left\lfloor{\frac{k}{2^{n-d}}}\right\rfloor$ , while for $n \gt 2^{2^d k-k-d+1}$ we have $f(n,k,d)=n + 2^d k -d-2$ , and obtain asymptotic results between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.

Abstract The notion of cross-intersecting set pair system of size $m$ , $ (\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m )$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne \emptyset$ , was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $m\le\binom{a+b}{a}$ if $|A_i|\le a$ and $|B_i|\le b$ for each $i$ . Our central problem is to see how this bound changes with the additional condition $|A_i\cap B_j|=1$ for $i\ne j$ . Such a system is called $1$ -cross-intersecting. We show that these systems are related to perfect graphs, clique partitions of graphs, and finite geometries. We prove that their maximum size is at least $5^{n/2}$ for $n$ even, $a=b=n$ , equal to $\bigl (\lfloor \frac{n}{2}\rfloor +1\bigr )\bigl (\lceil \frac{n}{2}\rceil +1\bigr )$ if $a=2$ and $b=n\ge 4$ , at most $|\cup _{i=1}^m A_i|$ , asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ( $|A_i\cap A_j|\le 1$ for $i\ne j$ ), asymptotically ${1\over 2}n^2$ if $\{A_i\}$ and $\{B_i\}$ are both linear hypergraphs.

Abstract A conjecture of Alon, Krivelevich and Sudakov states that, for any graph $F$ , there is a constant $c_F \gt 0$ such that if $G$ is an $F$ -free graph of maximum degree $\Delta$ , then $\chi\!(G) \leqslant c_F \Delta/ \log\!\Delta$ . Alon, Krivelevich and Sudakov verified this conjecture for a class of graphs $F$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot and Sereni that if $G$ is $K_{t,t}$ -free, then $\chi\!(G) \leqslant (t + o(1)) \Delta/ \log\!\Delta$ as $\Delta \to \infty$ . We improve this bound to $(1+o(1)) \Delta/\log\!\Delta$ , making the constant factor independent of $t$ . We further extend our result to the DP-colouring setting (also known as correspondence colouring), introduced by Dvořák and Postle.

Abstract We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $\epsilon>0$ there exists M such that every triangle-free graph G has an $\epsilon$ -approximate homomorphism to a triangle-free graph F on at most M vertices (here an $\epsilon$ -approximate homomorphism is a map $V(G) \to V(F)$ where all but at most $\epsilon \left\lvert{V(G)}\right\rvert^2$ edges of G are mapped to edges of F). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $\epsilon^{-1}$ . We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.

Abstract Consider the following experiment: a deck with m copies of n different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of ‘feedback’ is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in n, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the $m=2$ case was open.

Abstract In this short note, we prove the following analog of the Kővári–Sós–Turán theorem for intersection graphs of boxes. If G is the intersection graph of n axis-parallel boxes in $${{\mathbb{R}}^d}$$ such that G contains no copy of Kt,t, then G has at most ctn( log n)2d+3 edges, where c = c(d)>0 only depends on d. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension. Using this approach, we also show that a construction of Basit, Chernikov, Starchenko, Tao and Tran of K2,2-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon, Basavaraju, Chandran, Mathew and Rajendraprasad. We show that there exist graphs of separation dimension 4 having superlinear number of edges.