Triangle-free graphs with the fewest independent sets

TL;DR

This work addresses the extremal problem of bounding the number of independent sets in triangle-free graphs with given vertex set and average degree $d$. It extends Shearer’s induction to the hard-core model partition function $Z_G(\lambda)$, achieving an explicit lower bound $\log Z_G(\lambda) \ge \frac{W(\lambda d)^2+2W(\lambda d)-W(2\lambda)^2-2W(2\lambda)}{2(d-2)}|V|$ for $\lambda\in[0,1]$, with the $\lambda=1$ special case yielding a bound on $\log i(G)$ that implies $\log i(G) \ge (1+o(1))\frac{(\log d)^2}{2d}|V|$ as $d\to\infty$. The method hinges on a carefully chosen non-increasing convex function $f_\lambda$ built from the Lambert $W$-function and a local, vertex-removal induction that mirrors Shearer’s original argument. The paper proves sharpness of the bound (up to several leading terms) via a probabilistic construction using $G(n,d/n)$ and analyzes the dominant contribution to $Z_G(\lambda)$, confirming the bound is tight in the random-graph regime and relating to occupancy-fraction phenomena. Extensions to larger $\lambda$ are discussed, including numerical evidence suggesting broader validity, and the results deepen the connection between independence structures in triangle-free graphs and hard-core model techniques with potential Ramsey-theoretic implications.

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, Komlós and Szemerédi (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 $d\to\infty$.

Figures (1)

• Figure 1: A comparison plot of the boundary curves for $\mathcal{F}_1(d)$ in Theorem \ref{['thm:main']} redux.

Theorems & Definitions (16)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof : Proof of Lemma \ref{['lem:mainlambda']}
• Lemma 4
• proof
• Theorem 5
• Remark 6
• Remark 7