Combinatorial Structures on van der Waerden sets. (9th January 2015) - TYROS, KONSTANTINOS
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 subset A of a homogeneous tree T such that $\frac{|A\cap T(n)|}{|T(n)|}\geqslantδ, $ where T ( n ) denotes the n th level of T, for all n in 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 ∈ℕ of positive integers and for every real 0 < δ ⩽ 1, there exists a sequence ( nq ) q ∈ℕ of positive integers such that for every D ⊆ ∪ k ∏ q =0 k-1 [ nq ] satisfying $\frac{\big|D\cap \prod_{q=0}^{k-1} [n_q]\big|s}{\prod_{q=0}^{k-1}n_q}\geqslantδ$ for every k in a van der Waerden set, there is a sequence ( Jq ) q ∈ℕ, where Jq is an arithmetic progression of length mq contained in [ nq ] for all q, such that ∏ q =0 k-1 Jq ⊆ D for every k in a van der Waerden set. Moreover, working in an abstract setting, we may require Jq to be any configuration of natural numbers that can be found in an arbitrary set of positive density.