A rank lower bound for cutting planes proofs of Ramsey Theorem

Ramsey Theorem is a cornerstone of combinatorics and logic. In its simplest formulation it says that there is a function r such that any simple graph with r(k, s) vertices contains either a clique of size k or an independent set of size s. We study the complexity of proving upper bounds for the number r(k, k). In particular we focus on the propositional proof system cutting planes; we prove that the upper bound “r(k, k) ≤ 4” requires cutting planes proof of high rank. In order to do that we show a protection lemma which could be of independent interest. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems