@article{3064341,
    title = "A density version of the Halpern-Läuchli theorem",
    author = "Dodos, P. and Kanellopoulos, V. and Karagiannis, N.",
    journal = "Advances in Mathematics: Scientific Journal",
    year = "2013",
    volume = "244",
    pages = "955-978",
    doi = "10.1016/j.aim.2013.06.010",
    abstract = "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 ≥ 2 such that every t ∈ T has exactly b immediate successors. We show that for every d ≥ 1 and every tuple (T1., T d) of homogeneous trees, if D is a subset of the level product of (T1., T d) satisfying lim supn→∞|D∩(T1(n)×⋯×Td(n))||T1(n)×⋯×Td(n)|>0 then there exist strong subtrees (S1., S d) of (T1., T d) having a common level set such that the level product of (S1., S d) is a subset of D. © 2013 The Authors."
}