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

Polish space partition principles and the Halpern-L\"auchli theorem