The relativized weak pigeonhole principle states that if at least 2n out of n^2 pigeons fly into n holes, then some hole must be doubly occupied. We prove that every DNF-refutation of the CNF encoding of this principle requires size 2^{(\log n)^{3/2-\epsilon}} for every \epsilon > 0 and every sufficiently large n. By reducing it to the standard weak pigeonhole principle with 2n pigeons and n holes, we also show that this lower bound is essentially tight in that there exist DNF-refutations of size 2^{(\log n)^{O(1)}} even in R(\log). For the lower bound proof we need to discuss the existence of unbalanced low-degree bipartite expanders satisfying a certain robustness condition.