Choosability with Separation in Complete Multipartite Graphs

We show that there is a constant $k$ such that when $r \geq 2$ and $m \geq r^k$, the complete $r$-partite graph $K_{m*r}$ has a non-colorable list assignment $L$ such that $|L(v)| \geq \frac{7}{750}r\ln m$ for all $v$ and such that $|L(u) \cap L(v)| \leq \left\lfloor \frac{2r}{r-1} \right\rfloor$ whenever $u \neq v$. This roughly extends a result of Alon to the context of "choosability with separation", introduced by Kratochv\'il, Tuza, and Voigt.