Dense subsets of products of finite trees

We prove a “uniform” version of the finite density Halpern-L"{a}uchli 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 $b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_d\in \mathbb{N}$ with $b_i\geq 2$forall$i\in{1,…,d}$,everyinteger$k\meg 1$andeveryreal$0<\epsilon\leq 1$thereexistsaninteger$N$withthefollowingproperty\mathrm{.}If$(T_1,…,T_d)$arehomogeneoustreessuchthatthebranchingnumberof$T_i$is$b_i$forall$i\in{1,…,d}$,$L$isafinitesubsetof$$\mathbb{N}$$ of cardinality at least $N$ and $D$ is a subset of the level product of $(T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_d)$ satisfying $$\mathrm{\mid }D\cap (T_1(n)\times \mathrm{.}\mathrm{.}\mathrm{.}\times T_d(n))\mathrm{\mid }\ge ϵ\mathrm{\mid }{T}_1(n)\times \mathrm{.}\mathrm{.}\mathrm{.}\times T_d(n)\mathrm{\mid }$$ for every $n\in L$, then there exist strong subtrees $(S_1,\mathrm{.}\mathrm{.}\mathrm{.},S_d)$ of $(T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_d)$ of height $k$ and with common level set such that the level product of $(S_1,\mathrm{.}\mathrm{.}\mathrm{.},S_d)$ is contained in $D$. The least integer $N$ with this property will be denoted by $UDHL(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_d\mathrm{\mid }k,ϵ)$. 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 $UDHL(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_d\mathrm{\mid }k,ϵ)$.