A robust version of the multipartite Hajnal--Szemerédi theorem

In this note we show the following strengthening of a multipartite version of the Hajnal–Szemerédi theorem. For an integer $r\ge 3$ and $\gamma >0$, there exists a constant $C$ such that if $p\ge C{n}^{-2\mathrm{/}r}(\log n{)}^{1\mathrm{/}\left(\genfrac{}{}{0px}{}{r}{2}\right)}$ and $G$ is a balanced $r$-partite graph with each vertex class of size $n$ and ${\delta }^{\ast }(G)\ge (1-1\mathrm{/}r+\gamma )n$, then with high probability the random subgraph $G(p)$ of $G$ contains a $K_r$-factor. We also use it to derive corresponding transversal versions.