Formalizing Frankls 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.

💡 Research Summary

The paper presents a fully formalized, computer‑assisted approach to proving that a given set family is an FC‑family, a concept central to the long‑standing Frankl’s conjecture. Frankl’s conjecture asserts that in any finite family of sets closed under union there exists an element that belongs to at least half of the sets. An FC‑family is a subfamily with the property that any union‑closed family containing it automatically satisfies Frankl’s condition. Identifying FC‑families therefore prunes the search space for a general proof and has been a key strategy in many partial results.

The authors first encode finite sets and families of sets in the Isabelle/HOL proof assistant. A family is represented as a finite set of finite sets, and the union‑closed property is formalized as ∀A B ∈ F. A ∪ B ∈ F. The Frankl condition is expressed as a counting predicate: for each element x, the number of sets containing x must be at least |F|/2. Using Isabelle’s higher‑order logic, the definition of an FC‑family is introduced: a family F is an FC‑family iff for every union‑closed family G with F ⊆ G, G satisfies the Frankl condition. This definition is then proved equivalent to a collection of combinatorial lemmas that can be checked algorithmically.

The core contribution is a “proof‑by‑computation” pipeline that is itself verified inside Isabelle/HOL. Given a candidate family F, the system enumerates all possible supersets G (subject to a bound on the underlying universe and on the size of G) that are union‑closed and contain F. For each G the Frankl condition is tested. The enumeration, closure checking, and counting are implemented as executable Isabelle code via the code‑generation facility, producing SML/Scala programs that run under Isabelle’s trusted kernel. To accelerate the search, the authors translate the counting constraints into SAT/ILP instances; the solvers are invoked, and their results are imported back as Isabelle theorems, guaranteeing that no unchecked external computation contaminates the proof.

Using this verified pipeline, the authors re‑establish all FC‑families that have appeared in the literature (e.g., singletons {a}, doubletons {a,b}, the three‑element family {a,b},{a,c},{b,c}, etc.). Moreover, the search discovers a previously unknown FC‑family consisting of the four sets {a,b}, {a,c}, {b,c}, and {a,b,c}. The paper supplies the full Isabelle proof objects for this new family, demonstrating that any union‑closed family containing these four sets must have an element appearing in at least half of its members.

The significance of the work is twofold. First, it shows that sophisticated combinatorial searches can be fully integrated into a proof assistant, eliminating the traditional trust gap between human‑written proofs and computer‑generated data. Second, the discovery of a new FC‑family expands the catalog of known “building blocks” that can be used to attack Frankl’s conjecture, suggesting that further automated exploration may yield additional families and perhaps eventually a complete resolution. The authors also make all generated certificates and source code publicly available, ensuring reproducibility and providing a platform for future extensions to larger universes or more complex family structures.