Extending edge-colorings of complete hypergraphs into regular colorings

Let [GRAPHICS] be the collection of all h-subsets of an n-set. X superset of Y. Given a coloring (partition) of a set S subset of [GRAPHICS] , we are interested in finding conditions under which this coloring is extendible to a coloring of [GRAPHICS] so that the number of times each element of X appears in each color class (all sets of the same color) is the same number r. The case S = empty set, r = 1 was studied by Sylvester in the 18th century and remained open until the 1970s. The case h = 2, r = 1 is extensively studied in the literature and is closely related to completing partial symmetric Latin squares. For S = [GRAPHICS] , we settle the cases h = 4, vertical bar X vertical bar >= 4.847323 vertical bar Y vertical bar, and h = 5, vertical bar X vertical bar >= 6.285214 vertical bar Y vertical bar completely. Moreover, we make partial progress toward solving the case where s = [GRAPHICS] . 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.