Combinatorial Structures on van der Waerden sets

TL;DR

This paper extends classical combinatorial theorems to van der Waerden sets, establishing infinitary and density results about structured subsets within trees and Cartesian products.

Contribution

It provides an infinitary version of the Furstenberg-Weiss Theorem and demonstrates the existence of structured configurations in dense subsets over van der Waerden sets.

Findings

Every dense subset of a homogeneous tree contains a strong subtree with a van der Waerden level set.

Existence of structured product sets within dense subsets indexed by van der Waerden sets.

Generalization to arbitrary configurations in positive density sets.

Abstract

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.