Choosability with Separation in Complete Multipartite Graphs

We show that there is a constant $k$ such that when $r\ge 2$ and $m \geq r^k$,thecomplete$r$-partitegraph$K_{m*r}$ has a non-colorable list assignment $L$ such that $\mathrm{\mid }L(v)\mathrm{\mid }\ge \frac{7}{750}r\ln m$ for all $v$ and such that $\mathrm{\mid }L(u)\cap L(v)\mathrm{\mid }\le \lfloor \frac{2r}{r-1}\rfloor$ whenever $u\mathrm{\ne }v$. This roughly extends a result of Alon to the context of “choosability with separation”, introduced by Kratochv'il, Tuza, and Voigt.