No abstract

Abstract. We prove that the propositional translations of the KneserLovász theorem have polynomial size extended Frege proofs and quasipolynomial size Frege proofs. We present a new counting-based combinatorial proof of the Kneser-Lovász theorem that avoids the topological arguments of prior proofs for all but finitely many cases for each k. We introduce a miniaturization of the octahedral Tucker lemma, called the truncated Tucker lemma: it is open whether its propositional translations have (quasi-)polynomial size Frege or extended Frege proofs.

Bonet, Buss, and Pitassi [2] looked for examples of tautologies that might be conjectured to provide exponential separations between the Frege and extended Frege proof systems. They found only a small number of examples other than partial consistency statements. The first type of examples were based on linear algebra, and included the Oddtown Theorem, the Graham-Pollack Theorem, the Fisher Inequality, and the Ray-Chaudhuri-Wilson Theorem. The remaining example was Frankl's Theorem on the trace of sets.The four principles based on linear algebra all have short extended Frege proofs using facts about determinants and eigenvalues. The same is true for the "AB=I ⇒ BA=I" tautologies about square matrices A and B over GF 2 that was subsequently suggested by S. Cook. Recently, Hrubes and Tzameret [10] showed

We prove that the propositional translations of the Kneser-Lovász theorem have polynomial size extended Frege proofs and quasi-polynomial size Frege proofs. We present a new counting-based combi-natorial proof of the Kneser-Lovász theorem that avoids the topological arguments of prior proofs for all but finitely many cases for each k. We introduce a miniaturization of the octahedral Tucker lemma, called the truncated Tucker lemma: it is open whether its propositional translations have (quasi-)polynomial size Frege or extended Frege proofs.