Disproving the Neighborhood Conjecture

We study the following Maker/Breaker game. Maker and Breaker take turns in choosing vertices from a given n-uniform hypergraph F, with Maker going first. Maker's goal is to completely occupy a hyperedge and Breaker tries to avoid this. Beck conjectures that if the maximum neighborhood size of F is at most 2^n-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 2^n-1/n where Maker has a winning strategy. Finally we show that each n-uniform hypergraph with maximum degree at most 2^n-2/(en) has a proper halving 2-coloring, which solves another open problem posed by Beck related to the Neighborhood Conjecture.