Chong Shangguan (Tel-Aviv University)
Consider the following hat guessing game:
nn players are placed on nn vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of qq 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 GG, its hat guessing number HG(G)HG(G) is the largest integer qq such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of qq possible colors.
In 2008, Butler et al. asked whether the hat guessing number of the complete bipartite graph Kn,nKn,n is at least some fixed positive (fractional) power of nn. We answer this question affirmatively, showing that for sufficiently large nn, the complete rr-partite graph Kn,…,nKn,…,n satisfies HG(Kn,…,n)=Ω(nr−1r−o(1))HG(Kn,…,n)=Ω(nr−1r−o(1)). Our guessing strategy can also be extended to show that HG(→Cn,…,n)=Ω(n1r−o(1))HG(C→n,…,n)=Ω(n1r−o(1)), where →Cn,…,nC→n,…,n is the blow-up of a directed rr-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. Among other results, it is shown that under certain conditions, the linear hat guessing number of Kn,nKn,n is at most 33, exhibiting a huge gap from the Ω(n12−o(1))Ω(n12−o(1)) (nonlinear) hat guessing number of this graph.
This talk is based on a joint work with Noga Alon, Omri Ben-Eliezer and Itzhak Tamo.