The hat guessing number of graphs 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 rmHG(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.  :115,2008, 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 rpartite graph K_n,ldots,n satisfies rmHG(K_n,ldots,n)Omega(nfracr1ro(1)). Our guessing strategy is based on a probabilistic construction and other combinatorial ideas, and can be extended to show that rmHG(vecC_n,ldots,n)Omega(nfrac1ro(1)), where vecC_n,ldots,n is the blowup of a directed rcycle, and where for directed graphs each player sees only the hat colors of his outneighbors.

PotentialBased Strategies for TicTacToe on the Integer Lattice with Numerous Directions We consider a tictactoe game played on the ddimensional integer lattice.  The game that we investigate is a Makerxe2x80x93Breaker version of tictactoe.  In a Makerxe2x80x93Breaker game, the first player, Maker, only tries to occupy a winning line and the second player, Breaker, only tries to stop Maker from occupying a winning line. We consider the bounded number of directions game, in which we designate a finite set of directionvectors cal S subset Bbb Zd which determines the set of winning lines.  We show, by using the Erdxc5x91sxe2x80x93Selfridge theorem and a modification of a theorem by Beck about games played on almostdisjoint hypergraphs, that for the special case when the coordinates of each directionvector are bounded, i.e., when cal S subset  vecv : vecv_infty leq k , Breaker can win this game if the length of each winning line is on the order of d2lg(dk) and d2lg(k), respectively.  In addition, we show that Maker can build winning lines of length up to (1o(1))dlg k  if cal S is the set of  all  directionvectors with coordinates bounded by k.  We also apply these methods to the nconsecutive lattice points game on the Nd board with (essentially) cal S  Bbb Zd, and we show that the phase transition from a win for Maker to a win for Breaker occurs at n (do(1))lg N.