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
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
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