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… Expand

44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.

2003

TLDR

This work proves near-optimal rank bounds for Cutting Planes and Lovasz-Schrijver proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas.Expand

Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science

1994

TLDR

The main result shows that a family of tautologies, introduced in this paper requires exponential-sized tree-like CP proofs, and it follows that tree- like CP cannot polynomially simulate Frege systems.Expand

It is shown that the finite Ramsey theorem as a Δ0 schema is provable in IΔ0+Ω1 and that propositional formulas expressing the infinite Ramsey theorem have polynomial-size bounded-depth Frege proofs.Expand

This cutting plane proof system for proving the unsatisfiability of propositional formulas in conjunctive normalform is defined, and is the only known system with provably superpolynomial proof size, but polynomial size proofs for the pigeonhole formulas.Expand

It is shown that if the degree of falsity of a Cutting Planes proof Π is bounded by d(n) ≤ n/2, this proof can be easily transformed into a resolution proof of length at most |∏| · (d( n)n−1)64d(n).Expand