Combinatorial Structures on van der Waerden sets

Abstract

In this paper we provide two results. The first one consists of an infinitary version of the Furstenberg–Weiss theorem. More precisely we show that every subsetAof a homogeneous treeTsuch that$\frac{|A\cap T(n)|}{|T(n)|}\geqslant\delta,$whereT(n) denotes thenth level ofT, for allnin a van der Waerden set, for some positive real δ, contains a strong subtree having a level set which forms a van der Waerden set.The second result is the following. For every sequence (mq)q∈ℕof positive integers and for every real 0 < δ ⩽ 1, there exists a sequence (nq)q∈ℕof positive integers such that for everyD⊆ ∪k∏q=0k-1[nq] satisfying$\frac{\big|D\cap \prod_{q=0}^{k-1} [n_q]\big|s}{\prod_{q=0}^{k-1}n_q}\geqslant\delta$for everykin a van der Waerden set, there is a sequence (Jq)q∈ℕ, whereJqis an arithmetic progression of lengthmqcontained in [nq] for allq, such that ∏q=0k-1Jq⊆Dfor everykin a van der Waerden set. Moreover, working in an abstract setting, we may requireJqto be any configuration of natural numbers that can be found in an arbitrary set of positive density.