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

0369652 - MÚ 2012 RIV DE eng J - Journal Article
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 0933-5846. E-ISSN 1432-0665
R&D Projects: GA AV ČR IAA100190902; GA MŠMT LC505
Institutional research plan: CEZ:AV0Z10190503
Keywords : proof complexity * Ramsey theorem * resolution
Subject RIV: BA - General Mathematics
Impact factor: 0.341, year: 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.
Permanent Link: http://hdl.handle.net/11104/0203666