  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.
