Which patterns must a two-colouring of $K_n$ contain if each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours? We show that when $\varepsilon \gt 1/4$ , $K_n$ must contain a complete subgraph on $\Omega (\log n)$ vertices where one of the colours forms a balanced complete bipartite graph.

When $\varepsilon \leq 1/4$ , this statement is no longer true, as evidenced by the following colouring $\chi$ of $K_n$ . Divide the vertex set into $4$ parts nearly equal in size as $V_1,V_2,V_3, V_4$ , and let the blue colour class consist of the edges between $(V_1,V_2)$ , $(V_2,V_3)$ , $(V_3,V_4)$ , and the edges contained inside $V_2$ and inside $V_3$ . Surprisingly, we find that this obstruction is unique in the following sense. Any two-colouring of $K_n$ in which each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours (with $\varepsilon \gt 0$ ) contains a vertex set $S$ of order $\Omega _{\varepsilon }(\log n)$ on which one colour class forms a balanced complete bipartite graph, or which has the same colouring as $\chi$ .

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.

We prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$ -uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to Rödl, Ruciński, and Szemerédi. For graphs, Chvátal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chvátal’s theorem was taken by Pósa, who improved on Dirac’s tight minimum degree condition for Hamiltonian cycles by showing that a certain weaker condition on the degree sequence of a graph already yields a Hamiltonian cycle.

In this work, we take a similar step towards a full characterisation of all pair degree matrices that ensure the existence of tight Hamiltonian cycles in $3$ -uniform hypergraphs by proving a $3$ -uniform analogue of Pósa’s result. In particular, our result strengthens the asymptotic version of the result by Rödl, Ruciński, and Szemerédi.

We extend the notion of universal graphs to a geometric setting. A geometric graph is universal for a class $\mathcal H$ of planar graphs if it contains an embedding, that is, a crossing-free drawing, of every graph in $\mathcal H$ . Our main result is that there exists a geometric graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that is universal for $n$ -vertex forests; this generalises a well-known result by Chung and Graham, which states that there exists an (abstract) graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that contains every $n$ -vertex forest as a subgraph. The upper bound of $O\!\left(n \log n\right)$ edges cannot be improved, even if more than $n$ vertices are allowed. We also prove that every $n$ -vertex convex geometric graph that is universal for $n$ -vertex outerplanar graphs has a near-quadratic number of edges, namely $\Omega _h(n^{2-1/h})$ , for every positive integer $h$ ; this almost matches the trivial $O(n^2)$ upper bound given by the $n$ -vertex complete convex geometric graph. Finally, we prove that there exists an $n$ -vertex convex geometric graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that is universal for $n$ -vertex caterpillars.

A chordal graph is a graph with no induced cycles of length at least $4$ . Let $f(n,m)$ be the maximal integer such that every graph with $n$ vertices and $m$ edges has a chordal subgraph with at least $f(n,m)$ edges. In 1985 Erdős and Laskar posed the problem of estimating $f(n,m)$ . In the late 1980s, Erdős, Gyárfás, Ordman and Zalcstein determined the value of $f(n,n^2/4+1)$ and made a conjecture on the value of $f(n,n^2/3+1)$ . In this paper we prove this conjecture and answer the question of Erdős and Laskar, determining $f(n,m)$ asymptotically for all $m$ and exactly for $m \leq n^2/3+1$ .

For a fixed infinite graph $H$ , we study the largest density of a monochromatic subgraph isomorphic to $H$ that can be found in every two-colouring of the edges of $K_{\mathbb{N}}$ . This is called the Ramsey upper density of $H$ and was introduced by Erdős and Galvin in a restricted setting, and by DeBiasio and McKenney in general. Recently [4], the Ramsey upper density of the infinite path was determined. Here, we find the value of this density for all locally finite graphs $H$ up to a factor of 2, answering a question of DeBiasio and McKenney. We also find the exact density for a wide class of bipartite graphs, including all locally finite forests. Our approach relates this problem to the solution of an optimisation problem for continuous functions. We show that, under certain conditions, the density depends only on the chromatic number of $H$ , the number of components of $H$ and the expansion ratio $|N(I)|/|I|$ of the independent sets of $H$ .

Given a graph $G$ and an integer $\ell \ge 2$ , we denote by $\alpha _{\ell }(G)$ the maximum size of a $K_{\ell }$ -free subset of vertices in $V(G)$ . A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$ -vertex graphs $G$ with $\alpha _{\ell }(G) = o(n)$ , which can be seen as a Ramsey–Turán variant of the celebrated Hajnal–Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$ -factors in $n$ -vertex graphs $G$ with $\alpha _\ell (G)=n^{1-o(1)}$ for all $r\ge \ell \ge 2$ .

A random two-cell embedding of a given graph $G$ is obtained by choosing a random local rotation around every vertex. We analyse the expected number of faces of such an embedding, which is equivalent to studying its average genus. In 1991, Stahl [5] proved that the expected number of faces in a random embedding of an arbitrary graph of order $n$ is at most $n\log (n)$ . While there are many families of graphs whose expected number of faces is $\Theta (n)$ , none are known where the expected number would be super-linear. This led the authors of [1] to conjecture that there is a linear upper bound. In this note we confirm their conjecture by proving that for any $n$ -vertex multigraph, the expected number of faces in a random two-cell embedding is at most $2n\log (2\mu )$ , where $\mu$ is the maximum edge-multiplicity. This bound is best possible up to a constant factor.

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.

For a $k$ -uniform hypergraph $\mathcal{H}$ on vertex set $\{1, \ldots, n\}$ we associate a particular signed incidence matrix $M(\mathcal{H})$ over the integers. For $\mathcal{H} \sim \mathcal{H}_k(n, p)$ an Erdős–Rényi random $k$ -uniform hypergraph, ${\mathrm{coker}}(M(\mathcal{H}))$ is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes we show that for $p = \omega (1/n^{k - 1})$ , ${\mathrm{coker}}(M(\mathcal{H}))$ is torsion-free.

Given partially ordered sets (posets) $(P, \leq _P\!)$ and $(P^{\prime}, \leq _{P^{\prime}}\!)$ , we say that $P^{\prime}$ contains a copy of $P$ if for some injective function $f\,:\, P\rightarrow P^{\prime}$ and for any $X, Y\in P$ , $X\leq _P Y$ if and only if $f(X)\leq _{P^{\prime}} f(Y)$ . For any posets $P$ and $Q$ , the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$ -dimensional Boolean lattice are coloured in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on a poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$ -dimensional Boolean lattice $Q_n$ , as $n$ grows large. We show a sharp jump in behaviour of this number as a function of $n$ depending on whether or not $P$ contains a copy of either a poset $V$ , that is a poset on elements $A, B, C$ such that $B\gt C$ , $A\gt C$ , and $A$ and $B$ incomparable, or a poset $\Lambda$ , its symmetric counterpart. Specifically, we prove that if $P$ contains a copy of $V$ or $\Lambda$ then $R(P, Q_n) \geq n +\frac{1}{15} \frac{n}{\log n}$ . Otherwise $R(P, Q_n) \leq n + c(P)$ for a constant $c(P)$ . This gives the first non-marginal improvement of a lower bound on poset Ramsey numbers and as a consequence gives $R(Q_2, Q_n) = n + \Theta \left(\frac{n}{\log n}\right)$ .

Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when $k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of $\Omega ((n/k) \log n)$ on the stationary cover time, holding for any $n$ -vertex graph $G$ and any $1 \leq k =o(n\log n )$ . Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.

For a random binary noncoalescing feedback shift register of width $n$ , with all $2^{2^{n-1}}$ possible feedback functions $f$ equally likely, the process of long cycle lengths, scaled by dividing by $N=2^n$ , converges in distribution to the same Poisson–Dirichlet limit as holds for random permutations in $\mathcal{S}_N$ , with all $N!$ possible permutations equally likely. Such behaviour was conjectured by Golomb, Welch and Goldstein in 1959.

Given a family $\mathcal{F}$ of bipartite graphs, the Zarankiewicz number $z(m,n,\mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $\mathcal{F}$ as a subgraph (such $G$ is called $\mathcal{F}$ -free). For $1\leq \beta \lt \alpha \lt 2$ , a family $\mathcal{F}$ of bipartite graphs is $(\alpha,\beta )$ -smooth if for some $\rho \gt 0$ and every $m\leq n$ , $z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$ . Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any $(\alpha,\beta )$ -smooth family $\mathcal{F}$ , there exists $k_0$ such that for all odd $k\geq k_0$ and sufficiently large $n$ , any $n$ -vertex $\mathcal{F}\cup \{C_k\}$ -free graph with minimum degree at least $\rho (\frac{2n}{5}+o(n))^{\alpha -1}$ is bipartite. In this paper, we strengthen their result by showing that for every real $\delta \gt 0$ , there exists $k_0$ such that for all odd $k\geq k_0$ and sufficiently large $n$ , any $n$ -vertex $\mathcal{F}\cup \{C_k\}$ -free graph with minimum degree at least $\delta n^{\alpha -1}$ is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families $\mathcal{F}$ consisting of the single graph $K_{s,t}$ when $t\gg s$ . We also prove an analogous result for $C_{2\ell }$ -free graphs for every $\ell \geq 2$ , which complements a result of Keevash, Sudakov and Verstraëte.