Boxes, extended boxes and sets of positive upper density in the Euclidean space

Polona Durcik,Vjekoslav Kovač

doi:10.1017/s0305004120000316

Abstract

AbstractWe prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

Highlights

Euclidean Ramsey theory typically seeks for a given pattern, such as vertices of a square, an arithmetic progression, etc., in a single partition class determined by an arbitrary coloring of the Euclidean space
Lyall and Magyar [13] initiated the consideration of product-type patterns
We have already explained how Theorem 2 can be derived from Theorem 3, so we give outlines of proofs of Theorems 1 and 3

Summary

Introduction

Euclidean Ramsey theory typically seeks for a given pattern, such as vertices of a square, an arithmetic progression, etc., in a single partition class determined by an arbitrary (or only measurable) coloring of the Euclidean space. Lyall and Magyar [13] initiated the consideration of product-type patterns They proved that, for fixed a1, a2 > 0, a positive density subset of Rd1 × Rd2 , d1, d2 2, contains vertices of a rectangle,. Lyall and Magyar [12] worked on the Euclidean embedding of all large dilates of a fixed distance graph Their results do not include Theorem 1, since the boxes (or even rectangles) are simultaneously “too rigid” and “too degenerate;” compare with the definition of a proper k-degenerate distance graph from [12].

Scheme of the proofs

Analytical results