Number of independent transversals in multipartite graphs

An independent transversal in a multipartite graph is an independent set that intersects each part in exactly one vertex. We show that for every even integer $r\ge 2$, there exist $c_r>0$ and $n_0$ such that every $r$-partite graph with parts of size $n\ge n_0$ and maximum degree at most $rn/(2r-2)-t$, where $t=o(n)$, contains at least $c_r t n^{r-1}$ independent transversals. This is best possible up to the value of $c_r$. Our result confirms a conjecture of Haxell and Szab\'o from 2006 and partially answers a question raised by Erd\H{o}s in 1972 and studied by Bollob\'as, Erd\H{o}s and Szemer\'edi in 1975. We also show that, given any integer $s\ge 2$ and even integer $r\ge 2$, there exist $c_{r,s}>0$ and $n_0$ such that every $r$-partite graph with parts of size $n\ge n_0$ and maximum degree at most $rn/(2r-2)- c_{r, s} n^{1-1/s}$ contains an independent set with exactly $s$ vertices in each part. This is best possible up to the value of $c_{r, s}$ if a widely believed conjecture for the Zarankiewicz number holds. Our result partially answers a question raised by Di Braccio and Illingworth recently.