Are there Hard Examples for Frege Systems?

It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of combinatorial tautologies that we consider seem to give at most a quasipolynomial speed-up of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl. It is shown that Bondy’s theorem and a version of the Kruskal-Katona theorem actually have polynomial-size Frege proofs. Bondy’s theorem is shown to have constant-depth, polynomial-size proofs in Frege+PHP, and to be equivalent in I∆0 to the pigeonhole principle.