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arXiv · 2503.10002

Triangle-free graphs with the fewest independent sets

Pjotr Buys·Jan van den Heuvel·Ross J. Kang
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Abstract

Given $d>0$ and a positive integer $n$, let $G$ be a triangle-free graph on $n$ vertices with average degree $d$. With an elegant induction, Shearer (1983) tightened a seminal result of Ajtai, Koml\'os and Szemer\'edi (1980/1981) by proving that $G$ contains an independent set of size at least $(1+o(1))\frac{\log d}{d}n$ as $d\to\infty$. By a generalisation of Shearer's method, we prove that the number of independent sets in $G$ must be at least $\exp\left((1+o(1))\frac{(\log d)^2}{2d}n\right)$ as $d\to\infty$. This improves upon results of Cooper and Mubayi (2014) and Davies, Jenssen, Perkins, and Roberts (2018). Our method also provides good lower bounds on the independence polynomial of $G$, one of which implies Shearer's result itself. As certified by a classic probabilistic construction, our bound on the number of independent sets is sharp to several leading terms as $d\to\infty$.

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Pjotr Buys, Jan van den Heuvel, Ross J. Kang. 2025-03-13. Triangle-free graphs with the fewest independent sets. https://arxiv.org/abs/2503.10002

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