The missing log in large deviations for triangle counts

TL;DR

This paper provides precise large deviation estimates for the upper tail probability of triangle counts in Erdős-Rényi graphs, introducing a missing logarithmic factor in the exponent.

Contribution

It establishes the missing logarithmic factor in the large deviation exponent for triangle counts, potentially extending to general subgraph counts.

Findings

Sharp large deviation estimates with logarithmic correction

Method may extend to other subgraph counts

Completes the understanding of upper tail probabilities in random graphs

Abstract

This paper solves the problem of sharp large deviation estimates for the upper tail of the number of triangles in an Erdos-Renyi random graph, by establishing a logarithmic factor in the exponent that was missing till now. It is possible that the method of proof may extend to general subgraph counts.