Abstract
Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s < t let f s,t (n) = min{max{|S|: S ⊆ V (H) and H[S] contains no K s }}, where the minimum is taken over all K t -free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n 1/2+o(1)) ≤ f s,s+1(n) ≤ O(n 1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f s,s+1(n) ≤ O(n 2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n 1/2−ɛ) ≤ f s,s+k (n) ≤ O(n 1/2+ɛ. In addition, we also discuss some connections between the function f s,t and vertex Folkman numbers.
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Research partially supported by NSF grant DMS 0800070.
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Dudek, A., Rödl, V. On K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers. Combinatorica 31, 39–53 (2011). https://doi.org/10.1007/s00493-011-2626-3
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DOI: https://doi.org/10.1007/s00493-011-2626-3