Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs

TL;DR

This paper develops upper tail estimates for counting edges in random induced subhypergraphs and rooted random graphs, with applications to arithmetic progressions and Schur triples in random integer subsets.

Contribution

It introduces new upper tail bounds for hypergraph and rooted graph substructure counts using moment estimation techniques.

Findings

Upper tail bounds for hypergraph edge counts

Upper tail estimates for arithmetic progressions and Schur triples

Results applicable to random subsets of integers

Abstract

General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph H, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.