# On the average size of independent sets in triangle-free graphs

Davies, Ewan, Jenssen, Matthew, Perkins, Will and Roberts, Barnaby (2017) On the average size of independent sets in triangle-free graphs. Proceedings of the American Mathematical Society, 146 (1). pp. 111-124. ISSN 0002-9939

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Text - Accepted Version
Identification Number: 10.1090/proc/13728

## Abstract

We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on $n$ vertices with maximum degree $d$, showing that an independent set drawn uniformly at random from such a graph has expected size at least $(1+o_d(1)) \frac {\log d}{d}n$. This gives an alternative proof of Shearer's upper bound on the Ramsey number $R(3,k)$. We then prove that the total number of independent sets in a triangle-free graph with maximum degree $d$ is at least $\exp \left [\left (\frac {1}{2}+o_d(1) \right ) \frac {\log ^2 d}{d}n \right ]$. The constant $1/2$ in the exponent is best possible. In both cases, tightness is exhibited by a random $d$-regular graph. Both results come from considering the hard-core model from statistical physics: a random independent set $I$ drawn from a graph with probability proportional to $\lambda ^{\vert I\vert}$, for a fugacity parameter $\lambda >0$. We prove a general lower bound on the occupancy fraction (normalized expected size of the random independent set) of the hard-core model on triangle-free graphs of maximum degree $d$. The bound is asymptotically tight in $d$ for all $\lambda =O_d(1)$. We conclude by stating several conjectures on the relationship between the average and maximum size of an independent set in a triangle-free graph and give some consequences of these conjectures in Ramsey theory

Item Type: Article http://www.ams.org/publications/journals/journalsf... © 2017 American Mathematical Society Mathematics Q Science > QA Mathematics Departments > Mathematics 02 Feb 2018 17:11 20 Sep 2021 03:07 http://eprints.lse.ac.uk/id/eprint/86642