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We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d‐regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing that a union of copies of Kd,d maximizes the number of independent sets and the independence polynomial of a d‐regular graph. For matchings, this shows that the matching polynomial and the total number of matchings of a d‐regular graph are maximized by a union of copies of Kd,d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markström. In probabilistic language, our main theorems state that for all d‐regular graphs and all λ, the occupancy fraction of the hard‐core model and the edge occupancy fraction of the monomer‐dimer model with fugacity λ are maximized by Kd,d. Our method involves constrained optimization problems over distributions of random variables and applies to all d‐regular graphs directly, without a reduction to the bipartite case.

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 leastn. 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 expn . 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 λ |I| , for a fugacity parameter λ > 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 λ = 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.

We revisit Sapozhenko's classic proof on the asymptotics of the number of independent sets in the discrete hypercube {0, 1} d and Galvin's follow-up work on weighted independent sets. We combine Sapozhenko's graph container methods with the cluster expansion and abstract polymer models, two tools from statistical physics, to obtain considerably sharper asymptotics and detailed probabilistic information about the typical structure of (weighted) independent sets in the hypercube. These results refine those of Korshunov and Sapozhenko and Galvin, and answer several questions of Galvin. * University of Oxford, jenssen@maths.ox.ac.uk. † University of Illinois at Chicago, math@willperkins.org.

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