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