Disproving the Neighborhood Conjecture

TL;DR

This paper disproves Beck's Neighborhood Conjecture by constructing hypergraphs where Maker wins despite the maximum neighborhood size exceeding the conjectured bound, and also addresses related coloring problems.

Contribution

It provides explicit constructions of hypergraphs that counter the conjecture and solves an open problem on hypergraph coloring related to the conjecture.

Findings

Disproved Beck's Neighborhood Conjecture with explicit hypergraph constructions.

Established hypergraphs with maximum degree 2^(n-1)/n where Maker wins.

Proved existence of proper halving 2-colorings for hypergraphs with maximum degree up to 2^(n-2)/(en).

Abstract

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.