A concentration result with application to subgraph count

Abstract

Let H = (V,E) be a k-uniform hypergraph with a vertex set V and an edge set E. Let V_p be constructed by taking every vertex in V independently with probability p. Let X be the number of edges in E that are contained in V_p. We give a condition that guarantees the concentration of X within a small interval around its mean. The applicability of this result is demonstrated by deriving new sub-Gaussian tails for the number of copies of small complete and complete bipartite graphs in the binomial random graph, extending results of Ruci\'nski and Vu.