  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 ${\rm{HG}}(G)$ 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 $K_{n,\ldots,n}$ satisfies
${\rm{HG}}(K_{n,\ldots,n})=\Omega(n^{\frac{r-1}{r}-o(1)})$. Our guessing
strategy is based on a probabilistic construction and other combinatorial
ideas, and can be extended to show that
${\rm{HG}}(\vec{C}_{n,\ldots,n})=\Omega(n^{\frac{1}{r}-o(1)})$, where
$\vec{C}_{n,\ldots,n}$ is the blow-up of a directed $r$-cycle, and where for
directed graphs each player sees only the hat colors of his outneighbors.
