AbstractLet be the collection of all ‐subsets of an ‐set . Given a coloring (partition) of a set , we are interested in finding conditions under which this coloring is extendible to a coloring of so that the number of times each element of appears in each color class (all sets of the same color) is the same number . The case was studied by Sylvester in the 18th century and remained open until the 1970s. The case is extensively studied in the literature and is closely related to completing partial symmetric Latin squares. For , we settle the cases , and completely. Moreover, we make partial progress toward solving the case where . These results can be seen as extensions of the famous Baranyai’s theorem, and make progress toward settling a 40‐year‐old problem posed by Cameron.