ON THE AVERAGE SIZE OF INDEPENDENT SETS IN TRIANGLE-FREE GRAPHS

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)) 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 [(1/2 + o(d)(1)) log(2)d/d n . 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(vertical bar I vertical bar), 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.