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arXiv · 2410.18581

Counting sparse induced subgraphs in locally dense graphs

Rajko Nenadov
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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.

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Rajko Nenadov. 2024-10-24. Counting sparse induced subgraphs in locally dense graphs. https://arxiv.org/abs/2410.18581

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