Narrow Proofs May Be Maximally Long

We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n^Ω(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size n^O(w) is essentially tight. Moreover, our lower bounds can be generalized to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. Our results do not extend all the way to Lasserre, however-the formulas we study have Lasserre proofs of constant rank and size polynomial in both n and w.