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تسجيل مستخدم جديدOn the average size of independent sets in triangle-free graphs
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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 Shearers 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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