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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)

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> → (-1, 1) is a boolean-valued function on the discrete cube such that for any k > 0, [Σ<sub>|S| > k</sub> f(S)<sup>2</sup> < 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)

Let t ≥ 1 be an integer and let A be a family of subsets of {1, 2,. .. n} every two of which intersect in at least t elements. Identifying the sets with their characteristic vectors in {0, 1} n we study the maximal measure of such a family under a non uniform product measure. We prove, for a certain range of parameters, that the t-intersecting families of… (More)

We prove a new lower bound on the randomized decision tree complexity of monotone graph properties. For a monotone graph property $A$ of graphs on $n$ vertices, let $p=p(A)$ denote the threshold probability of $A$, namely the value of $p$ for which a random graph from $G(n,p)$ has property $A$ with probability $1/2$. Then the expected number of queries made… (More)

A family J of subsets of {1,. .. , n} is called a j-junta if there exists J ⊆ {1,. .. , n}, with |J| = j, such that the membership of a set S in J depends only on S ∩ J. In this paper we provide a simple description of intersecting families of sets. Let n and k be positive integers with k < n/2, and let A be a family of pairwise intersecting subsets of {1,.… (More)

We study thresholds for Ramsey properties of random discrete structures. In particular, we determine the threshold for Rado's theorem for solutions of partition regular systems of equations in random subsets of the integers and we prove the 1-statement of the conjectured threshold for Ram-sey's theorem for random hypergraphs. Those results were conjectured… (More)