Maker playing against an invisible Breaker

Abstract

We initiate the study of the phantom version of Maker-Breaker positional games. In a phantom game, the moves of one of the players are hidden from the other player, who still has the complete information. We look at the biased $(a:b)$ Maker-PhantomBreaker games where the board is the edge set of the complete graph on $n$ vertices, $K_n$, and Maker has no information about PhantomBreaker's choices of edges. We give randomized strategies for both players in four classical games: connectivity game, perfect matching game, mindegree-$k$ game and Hamiltonicity game. In particular, we focus on characterizing all biases $(a:b)$ for which Maker wins asymptotically almost surely.