List chromatic number of $G(n,r,s)$ graphs

We study the list chromatic number of generalized Johnson graphs. Recall that the vertices of such graphs are all possible $r$-element subsets of $[n]=\{1,\dots ,n\}$ and an edge is drawn between two vertices if and only if corresponding sets have exactly $s$ common elements. We prove that for any fixed $s\ge 1$ and $r\ge 2s+2$ we have ${\chi }_L(G(n,r,s))=\mathrm{\Theta }(n^{s+1}\ln n)$. In case where $s=0$ we prove that for all $4\ln n\le r\le n\mathrm{/}20$ we have ${\chi }_L(G(n,r,0))\ge \frac{n}{120}\ln \left(\frac{en}{r}\right)$. Together with previous results of Bulankina, Inozemtsev and Kupavskii this proves that ${\chi }_L(G(n,r,0))=\mathrm{\Theta }(n\ln \left(\frac{en}{r}\right))$ uniformly for $2\le r\le (1\mathrm{/}2-\epsilon )n$.