# Disproof of the Neighborhood Conjecture with Implications to SAT

## Abstract

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. Kratochvíl, 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). Kratochvíl, Savický and Tuza also gave the best known lower bound \(f(k) = \Omega\left(\frac{2^{k}}{k}\right)\), which is a consequence of the Lovász Local Lemma. We prove that, in fact, \(f(k) = \Theta\left(\frac{2^{k}}{k}\right)\), improving upon the best known upper bound \(O\left((\log k) \cdot \frac{2^{k}}{k}\right)\) 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 \(\cal{F}\), with Maker going first. Maker’s goal is to completely occupy a hyperedge and Breaker tries to prevent this. Beck conjectures that if the maximum neighborhood size of \(\cal{F}\) is smaller than 2^{n − 1}− 1 then Breaker has a winning strategy. We disprove this conjecture by establishing an *n*-uniform hypergraph with maximum neighborhood size 3 ·2 ^{n − 3} where Maker has a winning strategy. Moreover, we show how to construct an *n*-uniform hypergraph with maximum degree \(\frac{2^{n - 1}}{n}\) 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.

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