In 1975 Bollobás, Erdős, and Szemerédi asked the following question: given positive integers $n, t, r$ with $2\le t\le r-1$, what is the largest minimum degree $\delta (G)$ among all $r$-partite graphs $G$ with parts of size $n$ and which do not contain a copy of $K_{t+1}$? The $r=t+1$ case has attracted a lot of attention and was fully resolved by Haxell and Szabó, and Szabó and Tardos in 2006. In this article, we investigate the $r\gt t+1$ case of the problem, which has remained dormant for over 40 years. We resolve the problem exactly in the case when $r \equiv -1 \pmod{t}$, and up to an additive constant for many other cases, including when $r \geq (3t-1)(t-1)$. Our approach utilizes a connection to the related problem of determining the maximum of the minimum degrees among the family of balanced $r$-partite $rn$-vertex graphs of chromatic number at most $t$.

A hypergraph $\mathcal{F}$ is non-trivial intersecting if every pair of edges in it have a nonempty intersection, but no vertex is contained in all edges of $\mathcal{F}$. Mubayi and Verstraëte showed that for every $k \ge d+1 \ge 3$ and $n \ge (d+1)k/d$ every $k$-graph $\mathcal{H}$ on $n$ vertices without a non-trivial intersecting subgraph of size $d+1$ contains at most $\binom{n-1}{k-1}$ edges. They conjectured that the same conclusion holds for all $d \ge k \ge 4$ and sufficiently large $n$. We confirm their conjecture by proving a stronger statement.

They also conjectured that for $m \ge 4$ and sufficiently large $n$ the maximum size of a $3$-graph on $n$ vertices without a non-trivial intersecting subgraph of size $3m+1$ is achieved by certain Steiner triple systems. We give a construction with more edges showing that their conjecture is not true in general.

We study several parameters of a random Bienaymé–Galton–Watson tree $T_n$ of size $n$ defined in terms of an offspring distribution $\xi$ with mean $1$ and nonzero finite variance $\sigma ^2$. Let $f(s)=\mathbb{E}\{s^\xi \}$ be the generating function of the random variable $\xi$. We show that the independence number is in probability asymptotic to $qn$, where $q$ is the unique solution to $q = f(1-q)$. One of the many algorithms for finding the largest independent set of nodes uses a notion of repeated peeling away of all leaves and their parents. The number of rounds of peeling is shown to be in probability asymptotic to $\log n/\log (1/f'(1-q))$. Finally, we study a related parameter which we call the leaf-height. Also sometimes called the protection number, this is the maximal shortest path length between any node and a leaf in its subtree. If $p_1 = \mathbb{P}\{\xi =1\}\gt 0$, then we show that the maximum leaf-height over all nodes in $T_n$ is in probability asymptotic to $\log n/\log (1/p_1)$. If $p_1 = 0$ and $\kappa$ is the first integer $i\gt 1$ with $\mathbb{P}\{\xi =i\}\gt 0$, then the leaf-height is in probability asymptotic to $\log _\kappa \log n$.

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.

Let ${\mathbb{G}(n_1,n_2,m)}$ be a uniformly random m-edge subgraph of the complete bipartite graph ${K_{n_1,n_2}}$ with bipartition $(V_1, V_2)$, where $n_i = |V_i|$, $i=1,2$. Given a real number $p \in [0,1]$ such that $d_1 \,{:\!=}\, pn_2$ and $d_2 \,{:\!=}\, pn_1$ are integers, let $\mathbb{R}(n_1,n_2,p)$ be a random subgraph of ${K_{n_1,n_2}}$ with every vertex $v \in V_i$ of degree $d_i$, $i = 1, 2$. In this paper we determine sufficient conditions on $n_1,n_2,p$ and m under which one can embed ${\mathbb{G}(n_1,n_2,m)}$ into $\mathbb{R}(n_1,n_2,p)$ and vice versa with probability tending to 1. In particular, in the balanced case $n_1=n_2$, we show that if $p\gg\log n/n$ and $1 - p \gg \left(\log n/n \right)^{1/4}$, then for some $m\sim pn^2$, asymptotically almost surely one can embed ${\mathbb{G}(n_1,n_2,m)}$ into $\mathbb{R}(n_1,n_2,p)$, while for $p\gg\left(\log^{3} n/n\right)^{1/4}$ and $1-p\gg\log n/n$ the opposite embedding holds. As an extension, we confirm the Kim–Vu Sandwich Conjecture for degrees growing faster than $(n \log n)^{3/4}$.

We make the first steps towards generalising the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.

Hadwiger’s conjecture asserts that every graph without a $K_t$-minor is $(t-1)$-colourable. It is known that the exact version of Hadwiger’s conjecture does not extend to list colouring, but it has been conjectured by Kawarabayashi and Mohar (2007) that there exists a constant $c$ such that every graph with no $K_t$-minor has list chromatic number at most $ct$. More specifically, they also conjectured that this holds for $c=\frac{3}{2}$.

Refuting the latter conjecture, we show that the maximum list chromatic number of graphs with no $K_t$-minor is at least $(2-o(1))t$, and hence $c \ge 2$ in the above conjecture is necessary. This improves the previous best lower bound by Barát, Joret and Wood (2011), who proved that $c \ge \frac{4}{3}$. Our lower-bound examples are obtained via the probabilistic method.

For a subgraph $G$ of the blow-up of a graph $F$, we let $\delta ^*(G)$ be the smallest minimum degree over all of the bipartite subgraphs of $G$ induced by pairs of parts that correspond to edges of $F$. Johansson proved that if $G$ is a spanning subgraph of the blow-up of $C_3$ with parts of size $n$ and $\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$, then $G$ contains $n$ vertex disjoint triangles, and presented the following conjecture of Häggkvist. If $G$ is a spanning subgraph of the blow-up of $C_k$ with parts of size $n$ and $\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$, then $G$ contains $n$ vertex disjoint copies of $C_k$ such that each $C_k$ intersects each of the $k$ parts exactly once. A similar conjecture was also made by Fischer and the case $k=3$ was proved for large $n$ by Magyar and Martin.

In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of $G$ to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.