Extending Edge-colorings of Complete Hypergraphs into Regular Colorings
Amin Bahmanian

TL;DR
This paper investigates conditions for extending partial hypergraph colorings into regular colorings, providing new results for specific cases and advancing towards a long-standing open problem in combinatorics.
Contribution
It extends Baranyai's theorem to new cases of hypergraph colorings, addressing a 40-year-old problem and offering partial solutions for complex coloring extensions.
Findings
Complete solutions for h=4 with |X| ≥ 4.8473|Y| and h=5 with |X| ≥ 6.2852|Y|.
Partial progress on cases where S is the complement of a hypergraph.
Connections made to classical problems like Latin squares and longstanding conjectures.
Abstract
Let 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…
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