Frankl's Conjecture at Height Four and the Structure of Height-Five Counterexamples

Abstract

We study Frankl's union-closed sets conjecture through the height of the inclusion poset. Working in the equivalent empty-set-free formulation, where one seeks an element contained in strictly more than half of the members, we prove the conjecture for every finite union-closed family of height at most four. Equivalently, the usual at-least-half formulation holds for every union-closed family containing the empty set and having height at most five. We also develop a structural theory for the next unresolved case. Assuming a smallest empty-set-free counterexample of height at most five, we show that it has even cardinality $2t$, at least three critical elements of frequency $t$, and satisfies the minimal-counterexample bound $t \geq 2n-1$. Every critical element determines a coatom of the form $U \setminus \{x\}$, while every critical pair satisfies a dichotomy between a full double-avoidance top and a large avoidance fiber admitting a three-layer trace normal form. Coordinate deletion further yields an exact matching-defect obstruction. Finally, introducing the minimum number of join-irreducible members required to cover all critical elements, we exclude the five-cover case and show that this critical join-cover number is either three or four. These results substantially constrain any possible height-five counterexample while leaving the remaining transfer problem explicit.