Maker-Breaker Games on Random Hypergraphs

TL;DR

This paper analyzes Maker-Breaker games on random hypergraphs, identifying threshold probabilities for Maker's victory and classifying the nature of these thresholds as either local or global properties.

Contribution

It determines the threshold probabilities for Maker winning on random hypergraphs and characterizes the thresholds as either weak or semi-sharp based on hypergraph parameters.

Findings

Threshold probabilities for Maker's win are established.

Thresholds are classified as either local or global properties.

Conjecture that global thresholds are actually sharp.

Abstract

In this paper, we study Maker-Breaker games on the random hypergraph $H_{n,s,p}$, obtained from the complete $s$-graph by keeping every edge independently with probability $p$. We determine the threshold probability for the property of Maker winning the game as a function of $s$, the uniformity of the underlying hypergraph, as well as $m$, $b$, the number of vertices that Maker and Breaker are respectively allowed to pick each turn. In addition, we show that depending on those $m,b,s$, there are two types of thresholds: either being Maker-win is a local property and the threshold is weak, or it is related to global properties of the random hypergraph and the threshold is semi-sharp. We conjecture that in the latter case, the threshold is actually sharp.