Consider the following hat guessing game: n players are placed on n vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. Given a graph G, its hat guessing number $\mathrm{HG}(G)$ is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.

In 2008, Butler et al. asked whether the hat guessing number of the complete bipartite graph $K_{n,n}$ is at least some fixed positive (fractional) power of n. We answer this question affirmatively, showing that for sufficiently large n, the complete r-partite graph $K_{n,\dots ,n}$ satisfies $\mathrm{HG}(K_{n,\dots ,n})=\mathrm{\Omega }(n^{\frac{r-1}{r}-o(1)})$. Our guessing strategy is based on a probabilistic construction and other combinatorial ideas, and can be extended to show that $\mathrm{HG}({\overrightarrow{C}}_{n,\dots ,n})=\mathrm{\Omega }(n^{\frac{1}{r}-o(1)})$, where ${\overrightarrow{C}}_{n,\dots ,n}$ is the blow-up of a directed r-cycle, and where for directed graphs each player sees only the hat colors of his outneighbors.

Additionally, we consider related problems like the relation between the hat guessing number and other graph parameters, and the linear hat guessing number, where the players are only allowed to use affine linear guessing strategies. Several nonexistence results are obtained by using well-known combinatorial tools, including the Lovász Local Lemma and the Combinatorial Nullstellensatz. Among other results, it is shown that under certain conditions, the linear hat guessing number of $K_{n,n}$ is at most 3, exhibiting a huge gap from the $\mathrm{\Omega }(n^{\frac{1}{2}-o(1)})$ (nonlinear) hat guessing number of this graph.

考虑以下帽子猜谜游戏：n个玩家被放置在图形的n个顶点上，每个顶点都戴着帽子，帽子的颜色是从q种可能的颜色中任意选择的。每个玩家都可以看到邻居的帽子颜色，但是看不到自己的帽子颜色。根据预定的猜测策略和他们看到的帽子颜色，要求所有玩家同时猜测自己的帽子颜色，其中不允许他们之间进行交流。给定图G，其帽子猜数$\mathrm{HG}（G）$是最大的整数q，因此存在一种猜测策略，可以保证对q种可能的颜色的任何帽子分配至少有一个正确的猜测。

在2008年，Butler等人。问是否完整的二部图的帽子猜数${ķ}_{ñ，ñ}$至少是n的某个固定正（分数）幂。我们肯定地回答这个问题，表明对于足够大的n，完整的r局部图${ķ}_{ñ，\dots ，ñ}$ 满足 $\mathrm{HG}（{ķ}_{ñ，\dots ，ñ}）=\mathrm{\Omega }（{ñ}^{\frac{\mathrm{[R}-\mathrm{1个}}{\mathrm{[R}}-Ø（\mathrm{1个}）}）$。我们的猜测策略基于概率构造和其他组合思想，并且可以扩展为表明$\mathrm{HG}（{\overrightarrow{C}}_{ñ，\dots ，ñ}）=\mathrm{\Omega }（{ñ}^{\frac{\mathrm{1个}}{\mathrm{[R}}-Ø（\mathrm{1个}）}）$，在哪里 ${\overrightarrow{C}}_{ñ，\dots ，ñ}$是有向r循环的爆炸，在有向图上，每个玩家都只能看到其邻居的帽子颜色。

此外，我们考虑了相关问题，例如帽子猜数和其他图形参数之间的关系以及线性帽子猜数，在这种情况下，仅允许玩家使用仿射线性猜想策略。通过使用众所周知的组合工具，包括LovászLocal Lemma和Combinatorial Nullstellensatz，可以获得一些不存在的结果。除其他结果外，它表明，在某些条件下，线性帽子的猜测数${ķ}_{ñ，ñ}$ 最多为3，与 $\mathrm{\Omega }（{ñ}^{\frac{\mathrm{1个}}{2}-Ø（\mathrm{1个}）}）$ 此图的（非线性）帽子猜测数。