On a Clique Game and the Erdős-Hajnal Problem on High-Chromatic High-Girth Subgraphs

For a fixed positive integer $k$, two players, $\textsf {Builder}$ and $\textsf {Chooser}$, alternate turns playing the following game on a dynamically changing graph that is initially empty. In each round, $\textsf {Builder}$ introduces a new vertex with edges to all previous vertices and then partitions the entire edge set into two subsets, after which $\textsf {Chooser}$ deletes one of the two. $\textsf {Builder}$ attempts to build a clique of size $k$, while $\textsf {Chooser}$ attempts to prevent that. We prove tower-type upper and lower bounds on how many rounds $\textsf {Builder}$ needs to guarantee a $k$-clique.