Dense subsets of products of finite trees

Abstract

We prove a "uniform" version of the finite density Halpern-Läuchli Theorem. Specifically, we say that a tree $T$ is homogeneous if it is uniquely rooted and there is an integer $b\ge 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. We show the following. For every integer $d\ge 1$, every $b_1,...,b_d\in \mathbb{N}$ with $b_i\ge 2$ for all $i\in \{1,...,d\}$, every integer $k\meg 1$ and every real $0<\epsilon \le 1$ there exists an integer $N$ with the following property. If $(T_1,...,T_d)$ are homogeneous trees such that the branching number of $T_i$ is $b_i$ for all $i\in \{1,...,d\}$, $L$ is a finite subset of $\mathbb{N}$ of cardinality at least $N$ and $D$ is a subset of the level product of $(T_1,...,T_d)$ satisfying \[|D\cap \big(T_1(n)\times ...\times T_d(n)\big)| \geq ε|T_1(n)\times ...\times T_d(n)|\] for every $n\in L$, then there exist strong subtrees $(S_1,...,S_d)$ of $(T_1,...,T_d)$ of height $k$ and with common level set such that the level product of $(S_1,...,S_d)$ is contained in $D$. The least integer $N$ with this property will be denoted by $UDHL(b_1,...,b_d|k,\epsilon )$. The main point is that the result is independent of the position of the finite set $L$. The proof is based on a density increment strategy and gives explicit upper bounds for the numbers $UDHL(b_1,...,b_d|k,\epsilon )$.

arXiv comment36 pages, no figures; International Mathematics Research Notices, to appear