Name
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 vertical bar A boolean AND T(n)vertical bar/vertical bar T(n)vertical bar >= delta, where T(n) denotes the nth level of T, for all n in a van der Waerden set, for some positive real delta, contains a strong subtree having a level set which forms a van der Waerden set. The second result is the following. For every sequence (m(q))(q is an element of N) of positive integers and for every real 0 < delta <= 1, there exists a sequence (n(q))(q is an element of N) of positive integers such that for every D subset of boolean OR(k) Pi(k-1)(q=0)[n(q)] satisfying vertical bar D boolean AND Pi(k-1)(q=0)[n(q)]vertical bar s/Pi(k-1)(q=0)n(q) >= delta for every k in a van der Waerden set, there is a sequence (J(q))(q is an element of N,) where J(q) is an arithmetic progression of length m(q) contained in [n(q)] for all q, such that Pi(k-1)(q=0) J(q) subset of D for every k in a van der Waerden set. Moreover, working in an abstract setting, we may require J(q) to be any configuration of natural numbers that can be found in an arbitrary set of positive density. |
Year | DOI | Venue |
|---|---|---|
2015 | 10.1017/S0963548314000868 | COMBINATORICS PROBABILITY & COMPUTING |
Field | DocType | Volume |
Integer,Discrete mathematics,Combinatorics,Natural number,Homogeneous tree,Level set,Van der Waerden's theorem,Mathematics,Arithmetic progression | Journal | 24 |
Issue | ISSN | Citations |
6 | 0963-5483 | 0 |
PageRank | References | Authors |
0.34 | 5 | 1 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Konstantinos Tyros | 1 | 6 | 2.63 |