Counting sparse induced subgraphs in locally dense graphs

Abstract

An $n$-vertex graph $G$ is locally dense if every induced subgraph of size larger than $ζn$ has density at least $d > 0$, for some parameters $ζ, d > 0$. We show that the number of induced subgraphs of $G$ with $m$ vertices and maximum degree significantly smaller than $dm$ is roughly $\binom{ζn}{m}$, for $m \ll ζn$ which is not too small. This generalises a result of Kohayakawa, Lee, Rödl, 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 Erdős-Rogers function for graphs with small extremal number.

Theorems & Definitions (11)

• Lemma 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• proof : Proof of Lemma \ref{['lemma:count']}
• Theorem 3.1
• Lemma 3.2
• proof
• Theorem 3.3: Theorem 3 in mattheus24ramsey
• proof : Proof of Theorem \ref{['thm:no_sparse']}