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A note on propositional proof complexity of some Ramsey-type statements

  1. 1.
    0369652 - MU-W 2012 RIV DE eng J - Článek v odborném periodiku
    Krajíček, Jan
    A note on propositional proof complexity of some Ramsey-type statements.
    Archive for Mathematical Logic. Roč. 50, 1-2 (2011), s. 245-255 ISSN 1432-0665
    Grant CEP: GA AV ČR IAA100190902; GA MŠk LC505
    Výzkumný záměr: CEZ:AV0Z10190503
    Klíčová slova: proof complexity * Ramsey theorem * resolution
    Kód oboru RIV: BA - Obecná matematika
    Impakt faktor: 0.341, rok: 2011
    http://www.springerlink.com/content/q27255801x225772/

    A Ramsey statement denoted n -> (k)(2)(2) says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formulaRAM(n, k) of size O(n(k)) and with terms of size ((k)(2)). Let r(k) be the minimal n for which the statement holds. We prove that RAM(r(k), k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlak's work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4(k), k), but the proof complexity of these formulas in resolution R or in its extension R(log) is unknown. We define two relativizations of the Ramsey statement that still have quasi-polynomial size constant depth Frege proofs but for which we establish exponential lower bound for R.
    Trvalý link: http://hdl.handle.net/11104/0203666
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