Chain-making games in grid-like posets

TL;DR

This paper investigates Maker-Breaker and Walker-Blocker games on grid-like posets, establishing bounds on chain-building capabilities and connecting to Conway's Angel-Devil game for certain poset structures.

Contribution

It introduces new bounds for chain construction in grid-like posets and analyzes the Walker-Blocker game, linking it to Angel-Devil game solutions.

Findings

Maximum guaranteed chain size in product of chains is $k-loor{r/2}$.

Walker can guarantee a chain hitting all levels when $d \\geq 14$.

For $d=2$, Walker guarantees only 2/3 of the levels, which is asymptotically optimal.

Abstract

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 $k-\lfloor r/2\rfloor$, 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 {\it 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 $d\ge14$; 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, and 2/3 is asymptotically achievable in the product of two equal chains.