Combinatorial Structures on van der Waerden 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 $\frac{|A\cap T(n)|}{|T(n)|}\ge \delta$, 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 $$\begin{gathered} 0<\delta \\ leq1 \end{gathered}$$, 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 $$\frac{\big{|}D\cap \prod_{q=0}^{k-1} [n_q]\big{|}}{\prod_{q=0}^{k-1}n_q}\geqδ$$ 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.

24 pages