Fast winning strategies in a generalized van der Waerden game

TL;DR

This paper investigates the minimal number of moves Maker needs to win a generalized van der Waerden game, providing exact results for small sets and bounds for larger sets.

Contribution

It establishes precise winning move counts for sets of size up to 4 and shows the non-existence of winning strategies in minimal moves for larger sets.

Findings

Maker wins in |S| moves for |S| ≤ 3

Maker wins in 5 moves when |S|=4

No winning strategy in |S| moves for |S| ≥ 5

Abstract

Consider the following Maker-Breaker game. Fix a finite subset $S\subset\mathbb{N}$ of the naturals. The players Maker and Breaker take turns choosing previously unclaimed natural numbers. Maker wins by eventually building a homothetic copy $aS+b$ of $S$, where $a\in\mathbb{N}\setminus\{0\}$ and $b\in\mathbb{Z}$. This is a generalization of the van der Waerden game analyzed by Beck. By the Hales-Jewett theorem, there exists a constant $c$ depending only on $|S|$ such that Maker can win in $c$ or less moves. We show that Maker can win in $|S|$ moves if $|S|\leq 3$. When $|S|=4$, we show that Maker can always win in $5$ or less moves and describe all $S$ such that Maker can win in $4$ moves. If $|S|\geq 5$, Maker has no winning strategy in $|S|$ moves.