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 distinct 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. Kratochvíl, Savický and Tuza showed that for every k≥3 there is an integer f(k) such that every (k, f(k))-CNF formula is satisfiable, but (k, f(k) + 1)-SAT is already NP-complete (it is not known whether f(k) is computable). Kratochvíl, Savický and Tuza also gave the best known lower bound $f(k) = \Omega \left( {\tfrac{{2^k }} {k}} \right)$ , which is a consequence of the Lovász Local Lemma. We prove that, in fact, $f(k) = \Theta \left( {\tfrac{{2^k }} {k}} \right)$ , improving upon the best known upper bound $O\left( {(\log k) \cdot \tfrac{{2^k }} {k}} \right)$ by Hoory and Szeider. Finally we establish a connection between the class of trees we consider and a certain family of positional games. The Maker/Breaker game we study is as follows. Maker and Breaker take turns in choosing vertices from a given n-uniform hypergraph $\mathcal{F}$ , with Maker going first. Maker’s goal is to completely occupy a hyperedge and Breaker tries to prevent this. The maximum neighborhood size of a hypergraph $\mathcal{F}$ is the maximal s such that some hyperedge of $\mathcal{F}$ intersects exactly s other hyperedges. Beck conjectures that if the maximum neighborhood size of $\mathcal{F}$ is smaller than 2 ^n?1 ? 1 then Breaker has a winning strategy. We disprove this conjecture by establishing, for every n≥3, the existence of an n-uniform hypergraph with maximum neighborhood size 3·2 ^n?3 where Maker has a winning strategy. Moreover, we show how to construct, for every n, an n-uniform hypergraph with maximum degree at most $\frac{{2^{n + 2} }} {n}$ where Maker has a winning strategy. In addition we show that each n-uniform hypergraph with maximum degree at most $\frac{{2^{n - 2} }} {{en}}$ has a proper halving 2-coloring, which solves another open problem posed by Beck related to the Neighborhood Conjecture.