论文标题
莱文帽子问题的成功概率以及图中的独立集
The success probability in Levine's hat problem, and independent sets in graphs
论文作者
论文摘要
莱昂内尔·莱文（Lionel Levine）的帽子挑战赛有$ t $播放器，每个帽子都有（非常大或无限）的帽子，每个帽子都以随机的黑色或白色为独立着色。允许玩家在选择随机颜色之前进行协调，但不在之后。每个玩家都看到所有帽子，除了自己的头。然后，他们同时尝试并从各自的堆栈中挑选一顶黑色帽子。只有在他们都是正确的情况下，他们才被宣布成功。莱文的猜想是，当玩家数量增长时，成功概率趋于零。我们证明，这种成功概率严格降低了玩家的数量，并与图理论中的问题提出了一些联系：将图形中最大的独立集和其随机诱导的子图中的最大独立集的大小相关联，并界定了一组顶点的大小相交，使图中每个最大尺寸独立设置相交。
Lionel Levine's hat challenge has $t$ players, each with a (very large, or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to coordinate before the random colors are chosen, but not after. Each player sees all hats except for those on her own head. They then proceed to simultaneously try and each pick a black hat from their respective stacks. They are proclaimed successful only if they are all correct. Levine's conjecture is that the success probability tends to zero when the number of players grows. We prove that this success probability is strictly decreasing in the number of players, and present some connections to problems in graph theory: relating the size of the largest independent set in a graph and in a random induced subgraph of it, and bounding the size of a set of vertices intersecting every maximum-size independent set in a graph.
