# Fully Automatic, Verified Classification of all Frankl-Complete (FC(6)) Set Families

## TL;DR

This paper presents a fully automated and verified method for classifying all Frankl-Complete families over a six-element set, advancing the understanding of Frankl's conjecture through comprehensive enumeration and formal proof verification.

## Contribution

It provides the first complete classification of FC-families over a 6-element universe using automated, formally verified methods, extending previous partial results.

## Key findings

- All minimal FC-families over a 6-element set are characterized.
- All maximal non-FC families over a 6-element set are enumerated.
- The classification is formally verified within 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. A family Fc is called Frankl-complete (or FC-family) if in every union-closed family F containing Fc, one of the elements of union Fc occurs in at least half of the elements of F (so F satisfies the Frankl's condition). FC-families play an important role in attacking the Frankl's conjecture, since they enable significant search space pruning. We extend previous work by giving a total characterization of all FC-families over a 6-element universe, by defining and enumerating all minimal FC and maximal nonFC-families. We use a fully automated, computer assisted approach, formally verified within the proof-assistant Isabelle/HOL.