Proof complexity and the Lovasz-Kneser Theorem (I)

ABSTRACT. We investigate the proof complexity of a class of propositional formulas expressing a combinatorial principle known as the Kneser-Lovasz Theorem. This is a family of propositional tautologies, indexed by an nonnegative integer parameter k that generalizes the Pigeonhole Principle (PHP, 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 strengthening of the Kneser-Lovasz Theorem. On the other hand for the cases $k=2,3$ (for which combinatorial proofs of the Kneser-Lovasz Theorem are known) we give polynomial size Frege (k=2), respectively extended Frege (k=3) proofs. The paper concludes with a brief discussion of the results (that will be presented in a subsequent paper) on the complexity of the general case of the Kneser-Lovasz theorem.