On the choosability of complete multipartite graphs with part size three

Let Km∗r be the complete r-partite graph with m vertices in each part. Erdős, Rubin, and Taylor showed that K2∗r is r-choosable and suggested the problem of determining the choosability of Km∗r. We show that K3∗r is exactly ⌈(4r−1)/3⌉ choosable.