Algebraic Combinatorics Seminar - Jeremy Chizewer

Title: The Hat Guessing Number of Graphs

Abstract:  The hat guessing number HG(G) of a graph G on n vertices is defined in terms of the following game: n players are placed on the n vertices of G, 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. The hat guessing number 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 this talk, I will begin with an illustrative example and then show the lower bound on HG(G(n,1/2)), where G(n,1/2) denotes the random graph on n vertices where each edge is included uniformly and independently with probability 1/2. I will also discuss the linear hat guessing number. 

This is based on joint work with Noga Alon.