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.
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Received: May 11, 1998 Final version received: March 24, 1999
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Li, Y., Rousseau, C. & Zang, W. Asymptotic Upper Bounds for Ramsey Functions. Graphs Comb 17, 123–128 (2001). https://doi.org/10.1007/s003730170060
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DOI: https://doi.org/10.1007/s003730170060