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Furstenberg-Weiss have extended Szemer\'edi's theorem on arithmeticprogressions to trees by showing that a large subset of the tree containsarbitrarily long arithmetic subtrees. We study higher dimensional versions thatanalogously extend the multidimensional Szemer\'edi theorem by demonstratingthe existence of certain arithmetic structures in large subsets of a cartesianproduct of trees.

Arithmetic subtrees in large subsets of products of trees