A unified, modern treatment of the theory of random graphs-including recent results and
techniques Since its inception in the 1960s, the theory of random graphs has evolved into a …

We study thresholds for extremal properties of random discrete structures. We determine the
threshold for Szemerédi's theorem on arithmetic progressions in random subsets of the …

A remarkable lemma of Szemerédi asserts that, very roughly speaking, any dense graph can
be decomposed into a bounded number of pseudorandom bipartite graphs. This far …

In this survey we describe a recently-developed technique for bounding the number (and
controlling the typical structure) of finite objects with forbidden substructures. This technique …

For 0< γ≤ 1 and graphs G and H, write G→ γ H if any γ-proportion of the edges of G spans
at least one copy of H in G. As customary, write K r for the complete graph on r vertices. We …

One of the most basic questions one can ask about a graph H is: how many H-free graphs
on n vertices are there? For non-bipartite H, the answer to this question has been well …

We consider the binomial random graph Gp and determine a sharp threshold function for the
edge‐Ramsey property G_p → (C^ l_1, ..., C^ l_r) for all l1,…, lr, where Cl denotes the cycle …

A hereditary property of graphs is a collection of graphs which is closed under taking
induced subgraphs. The speed of P is the function n↦| Pn|, where Pn denotes the graphs of …

For graphs FF and HH, let f F, H (n) f _ F, H (n) be the minimum possible size of a maximum
FF‐free induced subgraph in an nn‐vertex HH‐free graph. This notion generalizes the …

A central problem in extremal graph theory is to estimate, for a given graph H, the number of
H-free graphs on a given set of n vertices. In the case when H is not bipartite, Erd̋s, Frankl …