Remarks on a Ramsey theory for trees

TL;DR

This paper explores a Ramsey theory for trees, extending classical results to colored binary trees and establishing bounds on monochromatic replicas, with implications for Szemerédi's theorem.

Contribution

It introduces a new quantitative bound for monochromatic tree replicas and provides a simplified combinatorial proof of a Furstenberg-Weiss theorem variant.

Findings

N(d,k) = Θ(dk log k) for monochromatic replica size

Density versions lead to short proofs of Furstenberg-Weiss theorem

Random split coloring algorithm effectively finds monochromatic structures

Abstract

Extending Furstenberg's ergodic theoretic proof for Szemer\'edi's theorem on arithmetic progressions, Furstenberg and Weiss (2003) proved the following qualitative result. For every d and k, there exists an integer N such that no matter how we color the vertices of a complete binary tree T_N of depth N with k colors, we can find a monochromatic replica of T_d in T_N such that (1) all vertices at the same level in T_d are mapped into vertices at the same level in T_N; (2) if a vertex x of T_d is mapped into a vertex y in T_N, then the two children of x are mapped into descendants of the the two children of y in T_N, respectively; and 3 the levels occupied by this replica form an arithmetic progression. This result and its density versions imply van der Waerden's and Szemer\'edi's theorems, and laid the foundations of a new Ramsey theory for trees. Using simple counting arguments and a randomized coloring algorithm called random split, we prove the following related result. Let N=N(d,k) denote the smallest positive integer such that no matter how we color the vertices of a complete binary tree T_N of depth N with k colors, we can find a monochromatic replica of T_d in T_N which satisfies properties (1) and (2) above. Then we have N(d,k)=\Theta(dk\log k). We also prove a density version of this result, which, combined with Szemer\'edi's theorem, provides a very short combinatorial proof of a quantitative version of the Furstenberg-Weiss theorem.