邻域猜想的反证
Disproving the Neighborhood Conjecture
摘要：
我们来看看下面的Maker/Breaker游戏。Maker和Breaker轮流从给定的n-一致超图F中选择顶点，Maker先去。Maker的目标是完全占据一个超边，而Breaker试图避免这一点。Beck指出，如果F的最大邻域大小至多为2^（n-1），则Breaker有获胜策略。我们通过建立一个最大邻域大小为3*2^（n-3）的n-一致超图来反驳这个猜想，其中Maker有一个获胜策略。此外，我们还讨论了如何构造一个最大度为2^（n-1）/n的n-一致超图，其中Maker有一个获胜策略.最后，我们证明了最大度不超过2^（n-2）/（en）的n-一致超图都有一个真二等分2-染色，从而解决了Beck提出的另一个与邻域猜想有关的公开问题.
英文摘要：
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.
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