# $H$-games played on vertex sets of random graphs

## TL;DR

This paper introduces a new class of positional games played on the vertex sets of random graphs, analyzing their thresholds and connections to vertex Ramsey properties, especially for cliques and cycles.

## Contribution

It extends the study of $H$-games to vertex sets in random graphs, establishing links with vertex Ramsey properties and highlighting unique behaviors for certain subgraphs.

## Key findings

- Threshold probabilities for $H$-games relate to vertex Ramsey properties.
- Games with $H$ as a triangle or forest behave differently from the general case.
- Connections between positional games and Ramsey theory are established.

## Abstract

We introduce a new type of positional games, played on a vertex set of a graph. Given a graph $G$, two players claim vertices of $G$, where the outcome of the game is determined by the subgraphs of $G$ induced by the vertices claimed by each player (or by one of them). We study classical positional games such as Maker-Breaker, Avoider-Enforcer, Waiter-Client and Client-Waiter games, where the board of the game is the vertex set of the binomial random graph $G\sim G(n,p)$. Under these settings, we consider those games where the target sets are the vertex sets of all graphs containing a copy of a fixed graph $H$, called $H$-games, and focus on those cases where $H$ is a clique or a cycle. We show that, similarly to the edge version of $H$-games, there is a strong connection between the threshold probability for these games and the one for the corresponding vertex Ramsey property (that is, the property that every $r$-vertex-coloring of $G(n,p)$ spans a monochromatic copy of $H$). Another similarity to the edge version of these games we demonstrate, is that the games in which $H$ is a triangle or a forest present a different behavior compared to the general case.