Proof Complexity and the Kneser-Lovász Theorem (I)

We investigate the proof complexity of a class of propositional formulas expressing a combinatorial principle known as the KneserLovász Theorem. This is a family of propositional tautologies, indexed by an nonnegative integer parameter k that generalizes the Pigeonhole Principle (obtained for k = 1). We show, for all fixed k, 2Ω(n) lower bounds on resolution complexity and exponential lower bounds for bounded depth Frege proofs. These results hold even for the more restricted class of formulas encoding Schrijver’s strenghtening of the Kneser-Lovász Theorem. On the other hand for the cases k = 2, 3 (for which combinatorial proofs of the Kneser-Lovász Theorem are known) we give polynomial size Frege (k = 2), respectively extended Frege (k = 3) proofs. The paper concludes with a brief announcement of the results (presented in subsequent work) on the complexity of the general case of the KneserLovász theorem.