Biased Maker‐Breaker games, introduced by Chvátal and Erdős, are central to the field of positional games and have deep connections to the theory of random structures. The main questions are to determine the smallest bias needed by Breaker to ensure that Maker ends up with an independent set in a given hypergraph. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies certain “container‐type” regularity conditions. This will enable us to answer the main question for hypergraph generalizations of the H‐building games studied by Bednarska and Łuczak as well as a generalization of the van der Waerden games introduced by Beck. We find it remarkable that a purely game‐theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, while the analogous questions about sparse random discrete structures are usually quite challenging.
We consider the strong Ramsey-type game R (k) (H, ℵ 0 ), played on the edge set of the infinite complete k-uniform hypergraph K k N . Two players, called FP (the first player) and SP (the second player), take turns claiming edges of K k N with the goal of building a copy of some finite predetermined k-uniform hypergraph H. The first player to build a copy of H wins. If no player has a strategy to ensure his win in finitely many moves, then the game is declared a draw.In this paper, we construct a 5-uniform hypergraph H such that R (5) (H, ℵ 0 ) is a draw. This is in stark contrast to the corresponding finite game R (5) (H, n), played on the edge set of K 5 n . Indeed, using a classical game-theoretic argument known as strategy stealing and a Ramsey-type argument, one can show that for every k-uniform hypergraph G, there exists an integer n 0 such that FP has a winning strategy for R (k) (G, n) for every n ≥ n 0 .
In a (1 : q) Maker-Breaker game, one of the central questions is to find (or at least estimate) the maximal value of q that allows Maker to win the game. Based on the ideas of Bednarska and Luczak [1], who studied biased H-games, we prove general winning criteria for Maker and Breaker and a hypergraph generalization of their result. Furthermore, we study the biased version of a strong generalization of the van der Waerden games introduced by Beck [2] and apply our criteria to determine the threshold bias of these games up to constant factor. As in the result of [1], the random strategy for Maker is again the best known strategy.
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