Ramsey's Theorem in Bounded Arithmetic

We shall show that the finite Ramsey theorem as a Δ0 schema is provable in IΔ0+Ω1. As a consequence we get that propositional formulas expressing the finite Ramsey theorem have polynomial-size bounded-depth Frege proofs.

[1]  Jan Krajícek,et al.  Bounded Arithmetic and the Polynomial Hierarchy , 1991, Ann. Pure Appl. Log..

[2]  Miklós Ajtai,et al.  The complexity of the Pigeonhole Principle , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[3]  Seinosuke Toda.  On the computational power of PP and (+)P , 1989, 30th Annual Symposium on Foundations of Computer Science.

[4]  J. Paris,et al.  Counting $Δ_0$ sets , 1987 .

[5]  Stephen A. Cook,et al.  The Relative Efficiency of Propositional Proof Systems , 1979, Journal of Symbolic Logic.

[6]  Peter Frankl,et al.  Intersection theorems with geometric consequences , 1981, Comb..

[7]  Jeff B. Paris,et al.  Provability of the Pigeonhole Principle and the Existence of Infinitely Many Primes , 1988, J. Symb. Log..

[8]  Armin Haken,et al.  The Intractability of Resolution , 1985, Theor. Comput. Sci..