On the Size of Minimal Unsatisfiable Formulas

Abstract

An unsatisfiable formula is called minimal if it becomes satisfiable whenever any of its clauses are removed. We construct minimal unsatisfiable $k$-SAT formulas with $\Omega(n^k)$ clauses for $k \geq 3$, thereby negatively answering a question of Rosenfeld. This should be compared to the result of Lovász [Studia Scientiarum Mathematicarum Hungarica 11, 1974, p113-114] which asserts that a critically 3-chromatic $k$-uniform hypergraph can have at most ${n\choose k-1}$ edges.