On June 27, 2023, Chong Shangguan (上官冲) from Shandong University gave a talk at the Discrete Math Seminar on the maximum number of colors of hats so that at least one of the players can correctly guess his or her hat color while only seeing the colors of their neighbors on a graph. The title of his talk was “the hat guessing number of graphs.”
Consider the following hat guessing game: players are placed on vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of 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 , its hat guessing number is the largest integer such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of possible colors.
In 2008, Butler, Hajiaghayi, Kleinberg, and Leighton asked whether the hat guessing number of the complete bipartite graph is at least some fixed positive (fractional) power of . We answer this question affirmatively, showing that for sufficiently large , . Our guessing strategy is based on some ideas from coding theory and probabilistic method.
Based on a joint work with Noga Alon, Omri Ben-Eliezer, and Itzhak Tamo.
For fixed integers , and , let denote the maximum number of edges in an -vertex -uniform hypergraph in which the union of arbitrary distinct edges contains at least vertices. In 1973, Brown, Erdős and Sós initiated the study of the function and they proved that . We will survey the state-of-art results about the study of and , where and . Although these two functions have been extensively studied, many interesting questions remain open.
