A density version of the Halpern-Läuchli theorem

We prove a density version of the Halpern-Läuchli 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 $D$.

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