Chain-making games in grid-like posets
Daniel W. Cranston, William B. Kinnersley, Kevin G. Milans, Gregory J. Puleo, Douglas B. West · Journal of Combinatorics · 2012
We study the Maker-Breaker game on chains in a poset.In a chainproduct poset, the maximum size of a chain that Maker can guarantee building is kr/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 ≥ 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; 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 ≤ d ≤ 13.In the product of two chains of equal size, Walker can guarantee 2/3 of the levels asymptotically.