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

Published:14 June 2016Publication History
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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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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 8, Issue 4
        July 2016
        97 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/2956681
        Issue’s Table of Contents

        Copyright © 2016 ACM

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 14 June 2016
        • Accepted: 1 April 2016
        • Revised: 1 January 2016
        • Received: 1 July 2015
        Published in toct Volume 8, Issue 4

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