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Abstract. In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem.Let Vn(p) = {0, 1} n denote the Hamming space endowed with the probability measure µp defined by µpLet A be a monotone subset of Vn. We say that A is symmetric if there is a transitive permutation group Γ on {1, 2, . . . , n} such that A is invariant under Γ. Theorem. For every symmetric monotone A, if µp(A) > then µq( Graph propertiesA graph property is a property of graphs which depends only on their isomorphism class. Let P be a monotone graph property; that is, if a graph G satisfies P then every graph H on the same set of vertices, which contains G as a subgraph satisfies P as well. Examples of such properties are: G is connected, G is Hamiltonian, G contains a clique (=complete subgraph) of size t, G is not planar, the clique number of G is larger than that of its complement, the diameter of G is at most s, etc.For a property P of graphs with a fixed set of n vertices we will denote by µ p (P ) the probability that a random graph on n vertices with edge probability p satisfies P . The theory of random graphs was founded by Erdös and Rényi [8,4], and one of their significant discoveries was the existence of sharp thresholds for various graph properties; that is, the transition from a property being very unlikely to it being very likely is very swift. Many results on various aspects of this phenomenon have appeared since then. In what follows c 1 , c 2 , etc. are universal constants. Theorem 1.1. Let P be any monotone property of graphs on n vertices. If µ p (P ) > then µ q (P ) > 1 − for q = p + c 1 log(1/2 )/ log n.

A set of permutations I ⊂ S n I \subset S_n is said to be k k -intersecting if any two permutations in I I agree on at least k k points. We show that for any k ∈ N k \in \mathbb {N} , if n n is sufficiently large depending on k k , then the largest k k -intersecting subsets of S n S_n are cosets of stabilizers of k k points, proving a conjecture of Deza and Frankl. We also prove a similar result concerning k k -cross-intersecting subsets. Our proofs are based on eigenvalue techniques and the representation theory of the symmetric group.

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