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

Abstract

Any valid Ramsey statement n −→ (k) 2 2 can be encoded into a DNF formula RAM(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 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.The complexity of proving various Ramsey-type combinatorial statements is well studied in connection with Peano arithmetic or systems of second order arithmetic, or even with set theory. The foremost example is the ParisHarrington extension of finite Ramsey theorem, see [17].However, even sooner Krishnamurthy and Moll [15] proposed Ramsey theorem as a source of hard propositional tautologies (we discuss this more below). In this paper we continue the study of propositional complexity of Ramsey theorem, motivated by the following problem: *

The Complexity of Proving That a Graph Is Ramsey

“…Krishnamurthy and Moll [17] proved partial results on the complexity of proving the exact upper bound, and conjectured this formula to be hard in general. Krajíček later proved an exponential lower bound on the length of bounded depth Frege proofs of the same statement [16]. The upper bound r(k) ≤ 4 k has short proofs in a relatively weak fragment of sequent calculus, in which every formula in a proof has small constant depth [20], [16].…”

“…Krajíček later proved an exponential lower bound on the length of bounded depth Frege proofs of the same statement [16]. The upper bound r(k) ≤ 4 k has short proofs in a relatively weak fragment of sequent calculus, in which every formula in a proof has small constant depth [20], [16]. Recently Pudlák [21] has shown a lower bound on proofs of r(k) ≤ 4 k in resolution.…”

On the Proof Complexity of Paris-Harrington and Off-Diagonal Ramsey Tautologies

“…Note that all known results deal with the diagonal Ramsey theorem, where one forbids cliques and independent sets of the same size k. Krishnamurthy and Moll [1981] proved a r(k, k)/2 width lower bound in Resolution and an exponential lower bound for the Davis-Putnam procedure for RAM(r(k, k); k, k). Recently, Krajíček [2011] established an exponential size lower bound in Resolution for the same principle. Pudlák [1991] provided that the formula RAM(4 k ; k, k) has a proof in bounded-depth Frege system of size 2 k O(1) (note that such a proof is polynomial in the size of the Ramsey principle and is quasi-polynomial in the number of variables).…”

“…It has been sometimes proposed (e.g., by Clote [1995]) that propositional encoding of logically strong combinatorial principles could produce hard tautologies for propositional proof systems. Krajíček [2011] recently dismissed this idea as impracticable.…”

A lower bound on the size of resolution proofs of the Ramsey theorem

“…This tautology is provable in a bounded depth Frege system, see [7,4]. For this tautology, Krajíček proved an exponential lower bound on tree-like resolution proofs with conjunctions of logarithmic size, [3].…”