Extending Edge-colorings of Complete Hypergraphs into Regular Colorings

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 $\binom{X}{h}$ be the collection of all $h$-subsets of an $n$-set $X\supseteq Y$. Given a coloring (partition) of a set $S\subseteq \binom{X}{h}$, we are interested in finding conditions under which this coloring is extendible to a coloring of $\binom{X}{h}$ 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 $S=\varnothing, r=1$ 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 $S=\binom{Y}{h}$, we settle the cases $h=4, |X|\geq 4.847323|Y|$, and $h=5, |X|\geq 6.285214|Y|$ completely. Moreover, we make partial progress toward solving the case where $S=\binom{X}{h}\backslash \binom{Y}{h}$. 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.