Subsets of products of finite sets of positive upper density

Todorčević, Stevo; Tyros, Konstantinos

doi:10.1016/j.jcta.2012.07.010

Cited by 7 publications

(8 citation statements)

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“…Milliken further proved in [17] that the collection of strong trees forms, in modern terminology, a topological Ramsey space. Further variations and applications include ω many perfect trees in [15]; partitions of products in [2]; the density version in [5]; the dual version in [21]; canonical equivalence relations on finite strong trees in [22]; and applications to colorings of subsets of the rationals in [1] and [23], and to finite Ramsey degrees of the Rado graph in [18] which in turn was applied to show the Rado graph has the rainbow Ramsey property in [4], to name just a few.…”

Section: Introductionmentioning

confidence: 99%

The Halpern–läuchli Theorem at a Measurable Cardinal

Dobrinen¹,

Hathaway²

2017

J. symb. log.

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Several variants of the Halpern-Läuchli Theorem for trees of uncountable height are investigated. For κ weakly compact, we prove that the various statements are all equivalent. We show that the strong tree version holds for one tree on any infinite cardinal. For any finite d ≥ 2, we prove the consistency of the Halpern-Läuchli Theorem on d many normal κ-trees at a measurable cardinal κ, given the consistency of a κ + d-strong cardinal. This follows from a more general consistency result at measurable κ, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD.

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Section: Introductionmentioning

confidence: 99%

The Halpern–läuchli Theorem at a Measurable Cardinal

Dobrinen¹,

Hathaway²

2017

J. symb. log.

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“…The second result included in this work consists of an extension of the main theorem of [18], which established a density version of a Ramsey theoretic result proved in [3,17]. In particular, it was shown that, for every positive real ε and every sequence (m q ) q∈N of positive integers, there exists a sequence (n q ) q∈N of positive integers with the following property.…”

Section: Introductionmentioning

confidence: 90%

“…For instance, each I q can be chosen to be an arithmetic or a polynomial progression. The construction of the sequence (I q ) q∈N is effective, avoiding, in particular, compactness arguments as in [18].…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Structures on van der Waerden sets

Tyros¹

2015

Combinator. Probab. Comp.

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In this paper we provide two results. The first one consists an infinitary version of the Furstenberg-Weiss Theorem. More precisely we show that every subset $A$ of a homogeneous tree $T$ such that $\frac{|A\cap T(n)|}{|T(n)|}\geq\delta$, where T(n) denotes the $n$-th level of $T$, for all $n$ in a van der Waerden set, for some positive real $\delta$, contains a strong subtree having a level sets which forms a van der Waerden set. The second result is the following. For every sequence $(m_q)_{q}$ of positive integers and for every real $0<\delta\\leq1$, there exists a sequence $(n_q)_{q}$ of positive integers such that for every $D\subseteq \bigcup_k\prod_{q=0}^{k-1}[n_q]$ satisfying $$\frac{\big{|}D\cap \prod_{q=0}^{k-1} [n_q]\big{|}}{\prod_{q=0}^{k-1}n_q}\geq\delta$$ for every $k$ in a van der Waerden set, there is a sequence $(J_q)_{q}$, where $J_q$ is an arithmetic progression of length $m_q$ contained in $[n_q]$ for all $q$, such that $\prod_{q=0}^{k-1}J_q\subseteq D$ for every $k$ in a van der Waerden set. Moreover, working in an abstract setting, we obtain $J_q$ to be any configuration of natural numbers that can be found in an arbitrary set of positive density.Comment: 24 page

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“…An important result regarding partition relations for trees is the Halpern-Läuchli Theorem [5], which concerns partitions of finite products of infinite trees. The Halpern-Läuchli Theorem has produced many variations and generalizations, such as Milliken's theorem [7] on finite partitions of strong trees, countable colorings of perfect trees [6], and a dual version [11]. For a fuller discussion of the theorem and its variants, see [3] and the references therein.…”

Section: Introductionmentioning

confidence: 99%