Proof Complexity and the Kneser-Lov\'asz Theorem

Abstract

We investigate the proof complexity of a class of propositional formulas expressing a combinatorial principle known as the Kneser-Lov\'{a}sz 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 $k$, $2^{\Omega(n)}$ 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-Lov\'{a}sz Theorem. On the other hand for the cases $k=2,3$ (for which combinatorial proofs of the Kneser-Lov\'{a}sz 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 proof complexity of the general case of the Kneser-Lov\'{a}sz theorem.