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تسجيل مستخدم جديدSuperfilters, Ramsey theory, and van der Waerdens Theorem
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Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerdens Theorem. We establish several properties of superfilters, which generalize both Ramseys Theorem and its variant for ultrafilters on the natural numbers. We use them to confirm a conjecture of Kov{c}inac and Di Maio, which is a generalization of a Ramsey theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindmans 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.
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We prove a canonical polynomial Van der Waerdens Theorem. More precisely, we show the following. Let ${p_1(x),ldots,p_k(x)}$ be a set of polynomials such that $p_i(x)in mathbb{Z}[x]$ and $p_i(0)=0$, for every $iin {1,ldots,k}$. Then, in any colouring
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In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological
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Inspired by Ramseys theorem for pairs, Rival and Sands proved what we refer to as an inside/outside Ramsey theorem: every infinite graph $G$ contains an infinite subset $H$ such that every vertex of $G$ is adjacent to precisely none, one, or infinite
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