Counting sparse induced subgraphs in locally dense graphs

TL;DR

This paper investigates the enumeration of sparse induced subgraphs in locally dense graphs, revealing that their count aligns with binomial coefficients under certain conditions, and applies this to improve bounds on the Erdős-Rogers function.

Contribution

It generalizes existing results on independent sets to a broader class of sparse subgraphs in locally dense graphs, providing new enumeration techniques.

Findings

Number of sparse induced subgraphs approximates binomial coefficients

Extension of previous results on independent sets in locally dense graphs

Improved bounds on the Erdős-Rogers function for graphs with small extremal number

Abstract

An $n$-vertex graph $G$ is locally dense if every induced subgraph of size larger than $\zeta n$ has density at least $d > 0$, for some parameters $\zeta, d > 0$. We show that the number of induced subgraphs of $G$ with $m$ vertices and maximum degree significantly smaller than $dm$ is roughly $\binom{\zeta n}{m}$, for $m \ll \zeta n$ which is not too small. This generalises a result of Kohayakawa, Lee, R\"odl, and Samotij on the number of independent sets in locally dense graphs. As an application, we slightly improve a result of Balogh, Chen, and Luo on the generalised Erd\H{o}s-Rogers function for graphs with small extremal number.