A concentration result with application to subgraph count

TL;DR

This paper establishes a concentration inequality for the number of hypergraph edges in a random subset of vertices, with applications to tail bounds for subgraph counts in random graphs.

Contribution

It provides a new condition ensuring concentration of hypergraph edge counts, extending tail bounds for subgraph occurrences in random graphs.

Findings

Derived new sub-Gaussian tail bounds for small complete graphs.

Extended previous results of Ruciński and Vu on subgraph counts.

Provided a general concentration condition for hypergraph edge counts.

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.