Avoider-Enforcer star games

Abstract

In this paper, we study $(1:b)$ Avoider-Enforcer games played on the edge set of the complete graph on $n$ vertices. For every constant $k\ge 3$ we analyse the $k$-star game, where Avoider tries to avoid claiming $k$ edges incident to the same vertex. We analyse both versions of Avoider-Enforcer games -- the strict and the monotone -- and for each provide explicit winning strategies for both players. We determine the order of magnitude of the threshold biases $f_{\mathcal{F}}^{mon}$, $f_{\mathcal{F}}^{-}$ and $f_{\mathcal{F}}^{+}$, where $\mathcal{F}$ is the hypergraph of the game.