Formalizing Frankl's Conjecture: FC-families

TL;DR

This paper formalizes a computer-assisted method using Isabelle/HOL to verify FC-families related to Frankl's conjecture, confirming known families and discovering a new one to aid in attacking the conjecture.

Contribution

It introduces a formalized, verified approach for proving FC-families, enhancing the tools available for studying Frankl's conjecture.

Findings

Confirmed known FC-families in literature

Discovered a new FC-family

Validated the formalization with proof assistant Isabelle/HOL

Abstract

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.