# Games with finitely generated structures

## TL;DR

This paper explores a variant of the Banach-Mazur game using finitely generated structures, characterizes winning strategies, and introduces weak Fraisse classes to extend classical Fraisse theory.

## Contribution

It introduces the concept of weak Fraisse classes and connects them to the Banach-Mazur game, extending classical Fraisse theory to finitely generated structures.

## Key findings

- Characterization of winning strategies in the game.
- Introduction of weak Fraisse classes.
- Extension of Fraisse theory to finitely generated structures.

## Abstract

We study the abstract Banach-Mazur game played with finitely generated structures instead of open sets. We characterize the existence of winning strategies aiming at a single countably generated structure. We also introduce the concept of weak Fraisse classes, extending the classical Fraisse theory and revealing its relations to our Banach-Mazur game.