Disproof of the Neighborhood Conjecture with Implications to SAT

We study a special class of binary trees. Our results have implications on Maker/Breaker games and SAT: We disprove a conjecture of Beck on positional games and construct an unsatisfiable k-CNF formula with few occurrences per variable, thereby improving a previous result by Hoory and Szeider and showing that the bound obtained from the Lovász Local Lemma is tight up to a constant factor. A (k, s)-CNF formula is a boolean formula in conjunctive normal form where every clause contains exactly k literals and every variable occurs in at most s clauses. The (k, s)-SAT problem is the satisfiability problem restricted to (k, s)-CNF formulas. Kratochv́ıl, Savický and Tuza showed that for every k ≥ 3 there is an integer f(k) such that every (k, f(k))-formula is satisfiable, but (k, f(k) + 1)-SAT is already NP-complete (it is not known whether f(k) is computable). Kratochv́ıl, Savický and Tuza also gave the best known lower bound f(k) = Ω “ 2 k k ” , which is a consequence of the Lovász Local Lemma. We prove that, in fact, f(k) = Θ “ 2 k k ” , improving upon the best known upper bound