## 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* 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.

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## Additional information

An extended abstract appeared in Proc. 17th European Symposium on Algorithms (ESA) (2009)

Research is supported by the SNF Grant 200021-118001/1.

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Gebauer, H. Disproof of the neighborhood conjecture with implications to SAT.
*Combinatorica* **32, **573–587 (2012). https://doi.org/10.1007/s00493-012-2679-y

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### Mathematics Subject Classification (2000)

- 05C35
- 05C57
- 05C65
- 05D40