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    Some properties of Ramsey numbers

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    AbstractIn this paper, some properties of Ramsey numbers are studied, and the following results are presented. 1.(1) For any positive integers k1, k2, …, km l1, l2, …, lm (m > 1), we have r ∏i=1m ki + 1, ∏i=1m li + 1 ≥ ∏i=1m [ r (ki + 1,li + 1) − 1] + 1.2.(2) For any positive integers k1, k2, …, km, l1, l2, …, ln , we have r ∑i=1m ki + 1, ∑j=1n lj + 1 ≥ ∑i=1m∑j=1n r (ki + 1,lj + 1) − mn + 1. Based on the known results of Ramsey numbers, some results of upper bounds and lower bounds of Ramsey numbers can be directly derived by those properties

    Superfilters, Ramsey theory, and van der Waerden's Theorem

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    Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerden's Theorem. We establish several properties of superfilters, which generalize both Ramsey's Theorem and its variant for ultrafilters on the natural numbers. We use them to confirm a conjecture of Ko\v{c}inac and Di Maio, which is a generalization of a Ramsey theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindman's 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman-Rado-Sanders, Rado, and others, where the colored sets can be much smaller than the full set of natural numbers.Comment: Among other things, the results of this paper imply (using its one-dimensional version) a higher-dimensional version of the Green-Tao Theorem on arithmetic progressions in the primes. The bibliography is now update