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