We show that there are graphs with n vertices containing no K5,5 which have about n7/4 edges, thus proving that ex(n, K5,5) ≥ (1 + o(1))n7/4. This bound gives an asymptotic improvement to the known lower bounds on ex(n, Kt, s) for t = 5 when 5 ≤ s ≤ 12, and t = 6 when 6 ≤ s ≤ 8.

A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers; we call such a linear extension a natural extension. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of order-invariance: if we condition on the set of the bottom k elements of the natural extension, each feasible ordering among these k elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.

We define a variant of the crossing number for an embedding of a graph G into ℝ3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the rectilinear space crossing numbers of pseudo-random graphs.

A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b ≥ 2, called the branching number of T, such that every t ∈ T has exactly b immediate successors. We study the behaviour of measurable events in probability spaces indexed by homogeneous trees.

Precisely, we show that for every integer b ≥ 2 and every integer n ≥ 1 there exists an integer q(b,n) with the following property. If T is a homogeneous tree with branching number b and {At:t ∈ T} is a family of measurable events in a probability space (Ω,Σ,μ) satisfying μ(At)≥ϵ>0 for every t ∈ T, then for every 0<θ<ϵ there exists a strong subtree S of T of infinite height, such that for every finite subset F of S of cardinality n ≥ 1 we have In fact, we can take q(b,n)= ((2b−1)2n−1−1)·(2b−2)−1. A finite version of this result is also obtained.

Simple families of increasing trees were introduced by Bergeron, Flajolet and Salvy. They include random binary search trees, random recursive trees and random plane-oriented recursive trees (PORTs) as important special cases. In this paper, we investigate the number of subtrees of size k on the fringe of some classes of increasing trees, namely generalized PORTs and d-ary increasing trees. We use a complex-analytic method to derive precise expansions of mean value and variance as well as a central limit theorem for fixed k. Moreover, we propose an elementary approach to derive limit laws when k is growing with n. Our results have consequences for the occurrence of pattern sizes on the fringe of increasing trees.

More than forty years ago, Erdős conjectured that for any , every k-uniform hypergraph on n vertices without t disjoint edges has at most max edges. Although this appears to be a basic instance of the hypergraph Turán problem (with a t-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all . This improves upon the best previously known range , which dates back to the 1970s.

Let F3,3 be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vertex from abc and 2 vertices from xyz. We show that for all n ≥ 6, the maximum number of edges in an F3,3-free 3-graph on n vertices is . This sharpens results of Zhou [9] and of the second author and Rödl [7].

Let kr(n, δ) be the minimum number of r-cliques in graphs with n vertices and minimum degree at least δ. We evaluate kr(n, δ) for δ ≤ 4n/5 and some other cases. Moreover, we give a construction which we conjecture to give all extremal graphs (subject to certain conditions on n, δ and r).