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Two extensions of Ramsey’s theorem. (English) Zbl 1280.05083

Summary: Ramsey’s theorem, in the version of P. Erdős and G. Szekeres [Compos. Math. 2, 463–470 (1935; Zbl 0012.27010)], states that every 2-coloring of the edges of the complete graph on \(\{1,2,\ldots,n\}\) contains a monochromatic clique of order \((1/2)\log n\).
In this article, we consider two well-studied extensions of Ramsey’s theorem. Improving a result of V. Rödl [J. Comb. Theory, Ser. A 102, No. 1, 229–240 (2003; Zbl 1015.05091)], we show that there is a constant \(c>0\) such that every 2-coloring of the edges of the complete graph on \(\{2,3, \ldots, n\}\) contains a monochromatic clique \(S\) for which the sum of \(1/\log i\) over all vertices \(i\in S\) is at least \(c\log\log\log n\). This is tight up to the constant factor \(c\) and answers a question of P. Erdős from [Combinatorica 1, 25–42 (1981; Zbl 0486.05001)]. Motivated by a problem in model theory, Väänänen asked [J. Nešetřil and J. A. Väänänen, Commentat. Math. Univ. Carol. 37, No. 3, 433–443 (1996; Zbl 0881.05096)] whether for every \(k\) there is an \(n\) such that the following holds: for every permutation \({\pi}\) of \(\{1, \ldots,k-1\}\), every 2-coloring of the edges of the complete graph on \(\{1,2,\ldots,n\}\) contains a monochromatic clique \(a_1<\cdots< a_k\) with
\[ a_{\pi(1)+1}-a_{\pi(1)}>a_{\pi(2)+1}-a_{\pi(2)}>\cdots>a_{\pi(k-1)+1}-a_{\pi(k-1)}. \]
That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, P. Erdős et al. [Isr. J. Math. 102, 283–295 (1997; Zbl 0884.05092)] answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in \(k\). We make progress towards this conjecture, obtaining an upper bound on \(n\) which is exponential in a power of \(k\). This improves a result of S. Shelah [Algorithms Comb. 14, 240–246 (1997; Zbl 0864.05082)], who showed that \(n\) is at most double-exponential in \(k\).

MSC:

05C55 Generalized Ramsey theory
05D10 Ramsey theory
05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.)
05C15 Coloring of graphs and hypergraphs
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References:

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