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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(More)
Consider a function f : f0;1g n ! f0;1g. The sensitivity of a point v 2 f0;1g n is jfv 0 : f (v 0) 6 = f (v); dist(v; v 0) = 1gj, i.e. the number of neighbors of the point in the discrete cube on which the value of f diiers. The average sensitivity of f is the average of the sensitivity of all points in f0;1g n. (This can also be interpreted as the sum of(More)
We consider powers of regular graphs defined by the weak graph product and give a characterization of maximum-size independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of(More)
Let f k (n; p) denote the probability that the random graph G(n; p) is k-colorable. We show that for every k 3, there exists d k (n) such that for any > 0, lim n!1 f k (n; d k (n) ? n) = 1 and lim n!1 f k (n; d k (n) + n) = 0 : As a result we conclude that for any given value of n the the chromatic number of G(n; d=n) is concentrated in one value for all(More)
We study the following one-person game against a random graph: the Player's goal is to 2-colour a random sequence of edges e1, e2,. .. of a complete graph on n vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochro-matic triangle is created. The online version of the game requires that the Player should colour(More)
A theorem of Bourgain [4] on Fourier tails states that if f :(-1, 1)<sup>n</sup> &#8594; (-1, 1) is a boolean-valued function on the discrete cube such that for any k &gt; 0, [&#931;<sub>|S| &gt; k</sub> f(S)<sup>2</sup> &lt; k<sup>-1/2 + o(1)</sup>, ] then essentially, f depends on only 2<sup>O(k)</sup> coordinates. This and related theorems such as(More)
Let σ ∈ S k and τ ∈ S n be permutations. We say τ contains σ if there exist 1 ≤ x 1 < x 2 <. .. < x k ≤ n such that τ (x i) < τ (x j) if and only if σ(i) < σ(j). If τ does not contain σ we say τ avoids σ. Let F (n, σ) = |{τ ∈ S n | τ avoids σ}|. Stanley and Wilf conjectured that for any σ ∈ S k there exists a constant c = c(σ) such that F (n, σ) ≤ c n for(More)