AbstractA coloring (partition) of the collection of all ‐subsets of a set is ‐regular if the number of times each element of appears in each color class (all sets of the same color) is the same number . We are interested in finding the conditions under which a given ‐regular coloring of is extendible to an ‐regular coloring of for and . The case was solved by Cruse, and due to its connection to completing partial symmetric latin squares, many related problems are extensively studied in the literature, but very little is known for . The case was solved by Häggkvist and Hellgren, settling a conjecture of Brouwer and Baranyai. The cases and were solved by Rodger and Wantland, and Bahmanian and Newman, respectively. In this paper, we completely settle the cases and .