Memory in games and universal graphs

I will present recent characterisations of positionality and finite memory in infinite duration games by means of universal graphs. The goal is to derive properties of games with a given objective W by understanding the structure of graphs satisfying W (meaning that all path colorations belong to W). We will look at some examples, review the caracterization for positionality (LICS 2022) and discuss a recent extension to finite memory obtained jointly with Antonio Casares.