Pairing strategies for the Maker-Breaker game on the hypercube with subcubes as winning sets

TL;DR

This paper investigates the Maker-Breaker game on hypercube vertices with subcubes as winning sets, providing strategies for Breaker to win under certain conditions related to the hypercube dimension and subcube size.

Contribution

It introduces a pairing strategy for Breaker that guarantees victory in the hypercube Maker-Breaker game for specific dimensions and subcube sizes, extending previous understanding.

Findings

Breaker wins if n is a power of 4 and k ≥ n/4 + 1

Breaker has a winning pairing strategy for all n ≥ 3 if k ≥ ⌊3n/7⌋ + 1

The results connect hypercube structure with positional game strategies.

Abstract

We consider the Maker-Breaker positional game on the vertices of the $n$-dimensional hypercube $\{0,1\}^n$ with $k$-dimensional subcubes as winning sets. We describe a pairing strategy which allows Breaker to win if $n$ is a power of 4 and $k \ge n/4 +1$. Our results also imply that for all $n \geq 3$ there is a Breaker's win pairing strategy if $k \ge \left\lfloor\frac{3}{7}n\right\rfloor +1$.