Short proofs of the Kneser-Lovász coloring principle

We prove that propositional translations of the Kneser-Lovász theorem have polynomial size extended Frege proofs and quasi-polynomial size Frege proofs for all fixed values of k. We present a new counting-based combinatorial proof of the Kneser-Lovász theorem based on the Hilton-Milner theorem; this avoids the topological arguments of prior proofs for all but finitely many base cases. We introduce new “truncated Tucker lemma” principles, which are miniaturizations of the octahedral Tucker lemma. The truncated Tucker lemma implies the Kneser-Lovász theorem. We show that the k=1 case of the truncated Tucker lemma has polynomial size extended Frege proofs.