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# Journal of Combinatorics

## Volume 3 (2012)

### Number 4

### Chain-making games in grid-like posets

Pages: 633 – 649

DOI: http://dx.doi.org/10.4310/JOC.2012.v3.n4.a3

#### Authors

#### Abstract

We study the Maker-Breaker game on chains in a poset. In a chain-product poset, 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 poset and $r$ is the maximum size of a factor chain. We also study a variant where Maker must build a chain in increasing order, called the ordered chain game. Within the bottom $k$ levels of a product of $d$ chains of size at least $k$, Walker can guarantee a chain that hits all levels if $d\ge14$; 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; when $d=3$, Walker cannot guarantee all levels, as shown by Clarke, Finbow, Fitzpatrick, Messenger, and Nowakowski by studying a related game. It is unknown whether Walker can guarantee all levels when $4\le d\le 13$. In the product of two chains of equal size, Walker can guarantee $2/3$ of the levels asymptotically.