  The Halpern-L\"auchli theorem, a combinatorial result about trees, admits an
elegant proof due to Harrington using ideas from forcing. In an attempt to
distill the combinatorial essence of this proof, we isolate various partition
principles about products of perfect Polish spaces. These principles yield
straightforward proofs of the Halpern-L\"auchli theorem, and the same forcing
from Harrington's proof can force their consistency. We also show that these
principles are not ZFC theorems by showing that they put lower bounds on the
size of the continuum.
