  We investigate Maker-Breaker games on graphs of size $\aleph_1$ in which
Maker's goal is to build a copy of the host graph. We establish a firm
dependence of the outcome of the game on the axiomatic framework. Relating to
this, we prove that there is a winning strategy for Maker in the
$K_{\omega,\omega_1}$-game under ZFC+MA+$\neg$CH and a winning strategy for
Breaker under ZFC+CH. We prove a similar result for the $K_{\omega_1}$-game.
Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under
ZFC+CH again.
