Partial independent transversals in multipartite graphs

Given integers $r>d\ge 0$ and an $r$-partite graph, an independent $(r-d)$-transversal or $(r-d)$-IT is an independent set of size $r-d$ that intersects each part in at most one vertex. We show that every $r$-partite graph with maximum degree $\Delta$ and parts of size $n$ contains an $(r-d)$-IT if $n> 2\Delta (1-\frac{1}{q})$, provided $q= \lfloor \frac{r}{d+1}\rfloor\ge \frac{4r}{4d+5}$. This is tight when $q$ is even and extends a classical result of Haxell in the $d=0$ case. When $q= \lfloor \frac{r}{d+1} \rfloor\ge \frac{6r+6d+7}{6d+7}$ is odd, we show that $n> 2\Delta(1-\frac{1}{q-1})$ guarantees an $(r-d)$-IT in any $r$-partite graph. This is also tight and extends a result of Haxell and Szab\'o in the $d=0$ case. In addition, we show that $n> 5\Delta/4$ guarantees a $5$-IT in any $6$-partite graph and this bound is tight, answering a question of Lo, Treglown and Zhao.