Ehud Friedgut

Sharp thresholds of graph properties, and the -sat problem

Consider G(n, p) to be the probability space of random graphs on n vertices with edge probability p. We will be considering subsets of this space defined by monotone graph properties. A monotone graph property P is a property of graphs such that a) P is invariant under graph automorphisims. b) If graph H has property P , then so does any graph G having H as a subgraph. A monotone symmetric family of graphs is a family defined by such a property. One of the first observations made about random graphs by Erdos and Renyi in their seminal work on random graph theory [12] was the existence of threshold phenomena, the fact that for many interesting properties P , the probability of P appearing in G(n, p) exhibits a sharp increase at a certain critical value of the parameter p. Bollobas and Thomason proved the existence of threshold functions for all monotone set properties ([6]), and in [14] it is shown that this behavior is quite general, and that all monotone graph properties exhibit threshold behavior, i.e. the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval. More precise analysis of the size of the threshold interval is done in [7]. This threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics. One of the main questions that arises in studying phase transitions is: how “sharp” is the transition? For example, one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kCNF Boolean formula. Nati Linial, who introduced me to this problem, suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition, answering the question posed in [14] as to the relation between the length of the threshold interval and the value of the critical probability. In this paper we indeed introduce a simple condition and prove it is sufficient. Stated roughly, in the setting of random graphs, the main theorem states that if a property has a coarse threshold, then it can be approximated by the property of having certain given graphs as a subgraph. This condition can be applied in a more

Every monotone graph property has a sharp threshold

In their seminal work which initiated random graph theory Erdos and Renyi 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 μp( 1, 2, . . . , n) = pk · (1 − p)n−k, where k = 1 + 2 + · · · + n. Let 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(A) > 1− for q = p+ c1 log(1/2 )/ logn. (c1 is an absolute constant.)

Boolean Functions With Low Average Sensitivity Depend On Few Coordinates

. The sensitivity of a point is dist, i.e. the number of neighbors of the point in the discrete cube on which the value of differs. The average sensitivity of is the average of the sensitivity of all points in . (This can also be interpreted as the sum of the influences of the variables on , or as a measure of the edge boundary of the set which is the characteristic function of.) We show here that if the average sensitivity of is then can be approximated by a function depending on coordinates where is a constant depending only on the accuracy of the approximation but not on . We also present a more general version of this theorem, where the sensitivity is measured with respect to a product measure which is not the uniform measure on the cube.

Elections Can be Manipulated Often

The Gibbard-Satterthwaite theorem states that every non-trivial voting method among at least 3 alternatives can be strategically manipulated. We prove a quantitative version of the Gibbard-Satterthwaite theorem: a random manipulation by a single random voter will succeed with non-negligible probability for every neutral voting method among 3 alternatives that is far from being a dictatorship.

A sharp threshold for k-colorability

. ABSTRACT: Let k be a fixed integer and fn , p denote the probability that the random k . . graph Gn , p is k-colorable. We show that for k G 3, there exists dn such that for any k e ) 0, dn y e

Intersecting Families of Permutations

A set of permutations

Boolean functions whose Fourier transform is concentrated on the first two levels

I \subset S_n

On the number of copies of one hypergraph in another

is said to be {\em k-intersecting} if any two permutations in

Ramsey Games Against a One-Armed Bandit

I

BÜCHI COMPLEMENTATION MADE TIGHTER

agree on at least