Arithmetic subtrees in large subsets of products of trees

TL;DR

This paper extends Szemerédi's theorem to higher-dimensional arithmetic structures within large subsets of Cartesian products of trees, revealing new combinatorial properties of these complex structures.

Contribution

It introduces higher-dimensional analogs of arithmetic progressions in products of trees, broadening the understanding of combinatorial configurations in these structures.

Findings

Large subsets of product of trees contain complex arithmetic structures.

Extension of multidimensional Szemerédi theorem to tree products.

New combinatorial properties identified in high-dimensional tree structures.

Abstract

Furstenberg-Weiss have extended Szemer\'edi's theorem on arithmetic progressions to trees by showing that a large subset of the tree contains arbitrarily long arithmetic subtrees. We study higher dimensional versions that analogously extend the multidimensional Szemer\'edi theorem by demonstrating the existence of certain arithmetic structures in large subsets of a cartesian product of trees.