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Archive for Mathematical Logic

, Volume 50, Issue 1–2, pp 245–255 | Cite as

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

  • Jan KrajíčekEmail author
Original Paper

Abstract

A Ramsey statement denoted \({n \longrightarrow (k)^2_2}\) 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 formula RAM(n, k) of size O(n k ) and with terms of size \({\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}\) . 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 Pudlák’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.

Keywords

Proof complexity Ramsey theorem Resolution 

Mathematics Subject Classification (2000)

03F20 68Q15 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPrague 8The Czech Republic

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