Lower Bounds for DNF-refutations of a Relativized Weak Pigeonhole Principle

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-ε
) for every ϵ > 0 and every sufficiently large n. For its proof we need to discuss the existence of unbalanced low-degree bipartite expanders satisfying a certain robustness condition.