Asymptotic Upper Bounds for Ramsey Functions

Abstract.

 We show that for any graph G with N vertices and average degree d, if the average degree of any neighborhood induced subgraph is at most a, then the independence number of G is at least Nf _{a +1}(d), where f _{a +1}(d)=∫₀ ¹(((1−t)^{1/( a +1)})/(a+1+(d−a−1)t))dt. Based on this result, we prove that for any fixed k and l, there holds r(K _k + _l ,K _n )≤ (l+o(1))n ^k/(logn)^{k −1}. In particular, r(K _k , K _n )≤(1+o(1))n ^{k −1}/(log n)^{k −2}.