Chain-making games in grid-like posets

الأصل · EN

We study the Maker-Breaker game on the hypergraph of chains of fixed size in a poset. In a product of chains, the maximum size of a chain that Maker can guarantee building is k- r/2, where k is the maximum size of a chain in the product, and r is the maximum size of a factor chain. We also study a variant in which Maker must follow the chain in order, called the Walker-Blocker game. In the poset consisting of the bottom k levels of the product of d arbitrarily long chains, Walker can guarantee a chain that hits all levels if d≥14; this result uses a solution to Conway's Angel-Devil game. When d=2, the maximum that Walker can guarantee is only 2/3 of the levels, and 2/3 is asymptotically achievable in the product of two equal chains.