Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least ε n² edges have to be deleted from it to make it H-free. We show that in this case G contains at least f(ε,H) n^h copies of H. This is proved ...

We describe a simple randomized construction for generating pairs of hash functions h₁,h₂ from a universe U to ranges V = [m] = (0,1,...,m-1) and W = [m] so that for every key set S ⊆ U with n = |S| ≤ m/(1 + ε) the (random) bipartite (multi)graph with ...

This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n,k, our extractor uses a random seed of length O(log n) to ...

Motivated by (the rate of information loss or) the rate at which the entropy of an ergodic Markov chain relative to its stationary distribution decays to zero, we study modified versions of the standard logarithmic Sobolev inequality in the discrete ...

We improve rapid mixing results for the simple Glauber dynamics designed to generate a random k-coloring of a bounded-degree graph.Let G be a graph with maximum degree Δ = Ω(log n), and girth ≥ 5. We prove that if k > Α Δ, where Α ≈ 1.763 then Glauber ...