On the choice number of complete multipartite graphs with part size four

Let ch(G) denote the choice number of a graph GG, and let Ks∗kKs∗k be the complete kk-partite graph with ss vertices in each part. Erdős, Rubin, and Taylor showed that ch(K2∗k)=k, and suggested the problem of determining the choice number of Ks∗kKs∗k. The first author established ch(K3∗k)=⌈4k−13⌉. Here we prove ch(K4∗k)=⌈3k−12⌉.