A density version of the Halpern-L\"{a}uchli theorem

TL;DR

This paper proves a density version of the Halpern-L"auchli theorem, confirming a conjecture by R. Laver, and shows that dense subsets of level products in homogeneous trees contain structured subtrees.

Contribution

It establishes a density version of the Halpern-L"auchli theorem for homogeneous trees, solving a conjecture and extending combinatorial tree partition results.

Findings

Proved a density version of the Halpern-L"auchli theorem.

Confirmed R. Laver's conjecture on dense subsets in homogeneous trees.

Identified conditions under which dense level sets contain structured subtrees.

Abstract

We prove a density version of the Halpern-L\"{a}uchli Theorem. This settles in the affirmative a conjecture of R. Laver. Specifically, let us say that a tree $T$ is homogeneous if $T$ has a unique root and there exists an integer $b\meg 2$ such that every $t\in T$ has exactly $b$ immediate successors. We show that for every $d\meg 1$ and every tuple $(T_1,...,T_d)$ of homogeneous trees, if $D$ is a subset of the level product of $(T_1,...,T_d)$ satisfying \[ \limsup_{n\to\infty} \frac{|D\cap \big(T_1(n)\times ... \times T_d(n)\big)|}{|T_1(n)\times ... \times T_d(n)|}>0\] then there exist strong subtrees $(S_1, ..., S_d)$ of $(T_1,...,T_d)$ having common level set such that the level product of $(S_1,...,S_d)$ is a subset of $D$.