Dan Cranston

*Maker-breaker games: Buliding a large chain in a poset *

**Abstract**. In a *maker-breaker game*, we fix a base set *X* and a collection of winning subsets *F*. The players Maker and Breaker alternate choosing elements from *X* and Maker wins if he eventually chooses all the elements in some subset in* F*. Otherwise Breaker wins. (A good example to keep in mind is a modified version of Tic-Tac-Toe, where the first player wins if he gets 3-in-a-row and the second player wins if and only if she stops the first player from winning.)

We consider the problem when the base set *X* is the set of elements of a partially ordered set (poset) *P* and Maker's winning set* F* is the collection of chains in *P* of a given length. When the poset * P * is a product of chains, we determine precisely the maximum length chain in *P* that Maker can attain.

We also consider a variation of the problem where Maker must choose the winning elements of the chain in order. By using surprising connections with Conway's Angel/Devil game, we determine precise bounds for many cases of this variation, as well.

This is joint work with Bill Kinnersley, Kevin Milans, Greg Puleo, and Doug West.

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