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

摘要

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 prepositional proof system cutting planes; we show that any cutting plane proof of the upper bound "r(k, k) ≤ 4~k" requires high rank. In order to do that we show a protection lemma which could be of independent interest.