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Extending regular edge-colorings of complete hypergraphs. (English) Zbl 07202775
Summary: 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
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