Extending regular edge-colorings of complete hypergraphs.

*(English)*Zbl 07202775Summary: A coloring (partition) of the collection \(\binom{X}{h}\) of all \(h\)-subsets of a set \(X\) is \(r\)-regular if the number of times each element of \(X\) appears in each color class (all sets of the same color) is the same number \(r\). We are interested in finding the conditions under which a given \(r\)-regular coloring of \(\binom{X}{h}\) is extendible to an \(s\)-regular coloring of \(\binom{Y}{h}\) for \(s \geqslant r\) and \(Y \supsetneq X\). The case \(h = 2\), \(r = s = 1\) 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 \(h \geqslant 3\). The case \(r = s = 1\) was solved by Häggkvist and Hellgren, settling a conjecture of Brouwer and Baranyai. The cases \(h = 2\) and \(h = 3\) were solved by Rodger and Wantland, and Bahmanian and Newman, respectively. In this paper, we completely settle the cases \(h = 4 \), \(| Y | \geqslant 4 | X |\) and \(h = 5 \), \(| Y | \geqslant 5 | X |\).

##### MSC:

05C | Graph theory |