This paper addresses the following question for a given graph H: What is the minimum number f(H) such that every graph with average degree at least f(H) contains H as a minor? Due to connections with Hadwiger's conjecture, this question has been studied in depth when H is a complete graph. Kostochka and Thomason independently proved that $f(K_t)=ct\sqrt{\ln t}$ . More generally, Myers and Thomason determined f(H) when H has a super-linear number of edges. We focus on the case when H has a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that if H has t vertices and average degree d at least some absolute constant, then $f(H)\leq 3.895\sqrt{\ln d}\,t$ . Furthermore, motivated by the case when H has small average degree, we prove that if H has t vertices and q edges, then f(H) ⩽ t + 6.291q (where the coefficient of 1 in the t term is best possible).

Let h > w > 0 be two fixed integers. Let H be a random hypergraph whose hyperedges are all of cardinality h. To w-orient a hyperedge, we assign exactly w of its vertices positive signs with respect to the hyperedge, and the rest negative signs. A (w,k)-orientation of H consists of a w-orientation of all hyperedges of H, such that each vertex receives at most k positive signs from its incident hyperedges. When k is large enough, we determine the threshold of the existence of a (w,k)-orientation of a random hypergraph. The (w,k)-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, when h = 2 and w = 1, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran. This settled a conjecture of Karp and Saks.

In [1], the authors consider a random walk (Z n,1, . . ., Z n,K+1) ∈ ${\mathbb{Z}}$ K+1 with the constraint that each coordinate of the walk is at distance one from the following coordinate. A functional central limit theorem for the first coordinate is proved and the limit variance is explicited. In this paper, we study an extended version of this model by conditioning the extremal coordinates to be at some fixed distance at every time. We prove a functional central limit theorem for this random walk. Using combinatorial tools, we give a precise formula of the variance and compare it with that obtained in [1].

Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.

Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.

In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

Nešetřil and Ossona de Mendez introduced the notion of first-order convergence, which unifies the notions of convergence for sparse and dense graphs. They asked whether, if (G i )i∈ℕ is a sequence of graphs with M being their first-order limit and v is a vertex of M, then there exists a sequence (v i )i∈ℕ of vertices such that the graphs G i rooted at v i converge to M rooted at v. We show that this holds for almost all vertices v of M, and we give an example showing that the statement need not hold for all vertices.

The first open case of the Brown–Erdős–Sós conjecture is equivalent to the following: for every c > 0, there is a threshold n0 such that if a quasigroup has order n ⩾ n0, then for every subset S of triples of the form (a, b, ab) with |S| ⩾ cn2, there is a seven-element subset of the quasigroup which spans at least four triples of S. In this paper we prove the conjecture for finite groups.

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr -packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.

In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

In this paper we study in complete generality the family of two-state, deterministic, monotone, local, homogeneous cellular automata in $\mathbb{Z}$d with random initial configurations. Formally, we are given a set $\mathcal{U}$ = {X1,. . . , Xm} of finite subsets of $\mathbb{Z}$d \ {0}, and an initial set A0 ⊂ $\mathbb{Z}$d of ‘infected’ sites, which we take to be random according to the product measure with density p. At time t ∈ $\mathbb{N}$, the set of infected sites At is the union of At-1 and the set of all x ∈ $\mathbb{Z}$d such that x + X ∈ At-1 for some X ∈ $\mathcal{U}$. Our model may alternatively be thought of as bootstrap percolation on $\mathbb{Z}$d with arbitrary update rules, and for this reason we call it $\mathcal{U}$-bootstrap percolation.

In two dimensions, we give a classification of $\mathcal{U}$-bootstrap percolation models into three classes – supercritical, critical and subcritical – and we prove results about the phase transitions of all models belonging to the first two of these classes. More precisely, we show that the critical probability for percolation on ($\mathbb{Z}$/n$\mathbb{Z}$)2 is (log n)−Θ(1) for all models in the critical class, and that it is n−Θ(1) for all models in the supercritical class.

The results in this paper are the first of any kind on bootstrap percolation considered in this level of generality, and in particular they are the first that make no assumptions of symmetry. It is the hope of the authors that this work will initiate a new, unified theory of bootstrap percolation on $\mathbb{Z}$d.

A graph on n vertices is ε-far from a property $\mathcal{P}$ if one has to add or delete from it at least εn2 edges to get a graph satisfying $\mathcal{P}$. A graph property $\mathcal{P}$ is strongly testable if for every fixed ε > 0 it is possible to distinguish, with one-sided error, between graphs satisfying $\mathcal{P}$ and ones that are ε-far from $\mathcal{P}$ by inspecting the induced subgraph on a random subset of at most f(ε) vertices. A property is easily testable if it is strongly testable and the function f is polynomial in 1/ε, otherwise it is hard. We consider the problem of characterizing the easily testable graph properties, which is wide open, and obtain several results in its study. One of our main results shows that testing perfectness is hard. The proof shows that testing perfectness is at least as hard as testing triangle-freeness, which is hard. On the other hand, we show that being a cograph, or equivalently, induced P3-freeness where P3 is a path with 3 edges, is easily testable. This settles one of the two exceptional graphs, the other being C4 (and its complement), left open in the characterization by the first author and Shapira of graphs H for which induced H-freeness is easily testable. Our techniques yield a few additional related results, but the problem of characterizing all easily testable graph properties, or even that of formulating a plausible conjectured characterization, remains open.

Estimating numerically the spectral radius of a random walk on a non-amenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups). In the genus 2 surface group, it improves by an order of magnitude the previous best bound, due to Bartholdi.

We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the lazy cop number of the hypercube. By coupling the probabilistic method with a potential function argument, we improve on the existing lower bounds for the lazy cop number of hypercubes.

We study the expected value of the length Ln of the minimum spanning tree of the complete graph Kn when each edge e is given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that limn→∞$\mathbb{E}$(Ln) = ζ(3) and show that

$$\mathbb{E}(L_n)=\zeta(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}},$$