Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas

Abstract

Seymour ( Quart. J. Math. Oxford 25 (1974) , 303–312) proved that a minimal non 2-colorable hypergraph on n vertices has at least n edges. A related fact is that a minimal unsatisfiable CNF formula in n variables has at least n + 1 clauses (an unpublished result of M. Tarsi.) The link between the two results is shown; both are given infinite versions and proved using transversal theory (Seymour's original proof used linear algebra). For the proof of the first fact we give a strengthening of König's duality theorem, both in the finite and infinite cases. The structure of minimal unsatisfiable CNF formulas in n variables containing precisely n + 1 clauses is characterised, and this characterization is given a geometric interpretation.