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The classical Halpern-L\"auchli theorem states that for any finite coloringof a finite product of finitely branching perfect trees of height $\omega$,there exist strong subtrees sharing the same level set such that tuplesconsisting of elements lying on the same level get the same color. Relative tolarge cardinals, we establish the consistency of a tail cone version of theHalpern-L\"auchli theorem at large cardinal, which, roughly speaking, dealswith many colorings simultaneously and diagonally. Among other applications, wegeneralize a polarized partition relation on rational numbers due to Laver andGalvin to one on linear orders of larger saturation.

A tail cone version of the Halpern-L\"auchli theorem at a large cardinal