   ISSN 1088-6826(online) ISSN 0002-9939(print)

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

Authors: Ewan Davies, Matthew Jenssen, Will Perkins and Barnaby Roberts
Journal: Proc. Amer. Math. Soc. 146 (2018), 111-124
MSC (2010): Primary 05C69; Secondary 05D10, 05D40
DOI: https://doi.org/10.1090/proc/13728
Published electronically: July 27, 2017
MathSciNet review: 3723125
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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 $\lambda ^{|I|}$, 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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Ewan Davies
Affiliation: Department of Mathematics, London School of Economics, London WC2A 2AE, United Kingdom
Email: e.s.davies@lse.ac.uk

Matthew Jenssen
Affiliation: Department of Mathematics, London School of Economics, London WC2A 2AE, United Kingdom
MR Author ID: 1015306
ORCID: 0000-0003-0026-8501
Email: m.o.jenssen@lse.ac.uk

Will Perkins
Affiliation: School of Mathematics, University of Birmingham, Birmingham B15 2TT, United Kingdom
MR Author ID: 939443
Email: math@willperkins.org

Barnaby Roberts
Affiliation: Department of Mathematics, London School of Economics, London WC2A 2AE, United Kingdom
Email: b.j.roberts@lse.ac.uk

Keywords: Independent sets, hard-core model, Ramsey theory
Received by editor(s): June 8, 2016
Received by editor(s) in revised form: March 6, 2017
Published electronically: July 27, 2017
Additional Notes: The third author was supported in part by EPSRC grant EP/P009913/1.
Communicated by: Patricia Hersh