Formalizing Frankl’s Conjecture: FC-Families
The Frankl’s conjecture, formulated in 1979. and still open, states that in every family of sets closed for unions there is an element contained in at least half of the sets. FC-families are families for which it is proved that every union-closed family containing them satisfies the Frankl’s condition (e.g., in every union-closed family that contains a one-element set a, the element a is contained in at least half of the sets, so families of the form a are the simplest FC-families). FC-families play an important role in attacking the Frankl’s conjecture, since they enable significant search space pruning. We present a formalization of the computer assisted approach for proving that a family is an FC-family. Proof-by-computation paradigm is used and the proof assistant Isabelle/HOL is used both to check mathematical content, and to perform (verified) combinatorial searches on which the proofs rely. FC-families known in the literature are confirmed, and a new FC-family is discovered.
KeywordsWeight Function Java Program Proof Assistant Semimodular Lattice Combinatorial Search
Unable to display preview. Download preview PDF.
- 2.Appel, K.I., Haken, W.: Every Planar Map is Four Colorable. American Mathematical Society (1989)Google Scholar
- 3.Bošnjak, I., Marković, P.: The 11-element Case of Frankl’s Conjecture. Electronic Journal of Combinatorics 15(1) (2008)Google Scholar
- 5.Gao, W., Yu, H.: Note on the Union-Closed Sets Conjecture. Ars Combinatorica 49 (1998)Google Scholar
- 6.Gonthier, G.: Formal Proof – the Four-Color Theorem. Notices of AMS 55(11) (2008)Google Scholar
- 13.Robertson, N., Sanders, D.P., Seymour, P.D., Thomas, R.: The Four Colour Theorem. Journal of Combinatorial Theory, Series B (1997)Google Scholar
- 17.Živković, M., Vučković, B.: The 12-element Case of Frankls Conjecture (submitted, 2012)Google Scholar