Summary: It was known that free Abelian groups do not admit a Hausdorff compact group topology. M. G. Tkachenko showed in [Sov. Math. 34, No. 5, 79-86 (1990); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1990, No. 5(336), 68-75 (1990; Zbl 0714.22001)] that, under CH, a free Abelian group of size \({\mathfrak c}\) admits a Hausdorff countably compact group topology.
We show that no Hausdorff group topology on a free Abelian group makes its \(\omega \)-th power countably compact. In particular, a free Abelian group does not admit a Hausdorff \(p\)-compact nor a sequentially compact group topology. Under CH, we show that a free Abelian group does not admit a Hausdorff initially \(\omega _1\)-compact group topology. We also show that the existence of such a group topology is independent of \({\mathfrak c}= \aleph _2\).