Tree resolution proofs of the weak pigeon-hole principle

We prove that any optimal tree resolution proof of PHP/sub n//sup m/ is of size 2/sup /spl theta/(n log n)/, independently from m, even if it is infinity. So far, only a 2/sup /spl Omega/(n)/ lower bound has been known in the general case. We also show that any, not necessarily optimal, regular tree resolution proof PHP/sub n//sup m/ is bounded by 2/sup O(n log m)/. To the best of our knowledge, this is the first time the worst case proof complexity has been considered. Finally, we discuss possible connections of our result to Riis' (1999) complexity gap theorem for tree resolution.