A rank lower bound for cutting Planes Proofs of Ramsey's Theorem

Ramsey's Theorem is a cornerstone of combinatorics and logic. In its simplest formulation it says that for every k>0 and s>0, there is a minimum number r(k,s) such that any simple graph with at least 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 show that any cutting plane proof of the upper bound “r(k,k)≤4k” requires high rank. In order to do that we show a protection lemma which could be of independent interest.