A Rank Lower Bound for Cutting Planes Proofs of Ramsey’s Theorem

Abstract. Ramsey’s 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 prov-ing 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) ≤ 4k ” requires cutting planes proof of high rank. In order to do that we show a protection lemma which could be of independent interest.