Unsatisfiable Linear CNF Formulas Are Large, and Difficult to Construct Explicitely

We call a CNF formula linear if any two clauses have at most one variable in common. We show that there exist unsatisfiable linear k-CNF formulas with at most O(k4) clauses, and on the other hand, any linear k-CNF formula with at most 4 k 4e2k3 clauses is satisfiable. The upper bound uses a probabilistic construction, and we have no explicit construction coming even close to it. We give some arguments why it is difficult to find explicit constructions: First, any treelike resolution refutation of any unsatisfiable linear k-CNF formula has size at least 2 k−1 2 . Second, if we require the unsatisfiable linear k-CNF formula to exhibit a certain recursive structure, then we need at least α ...α clauses, where α is roughly 2 and the size of this tower is roughly k.