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 $k\ge 3$, thereby negatively answering a question of Rosenfeld. This should be compared to the result of Lovász which asserts that a critically 3-chromatic $k$-uniform hypergraph can have at most $\left(\genfrac{}{}{0ex}{}{n}{k-1}\right)$ edges.

4 pages