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完全超图的边染色到正则染色的推广

Extending Edge-colorings of Complete Hypergraphs into Regular Colorings

Amin Bahmanian

备注：
14 pages. arXiv admin note: text overlap with arXiv:2009.09674
发表信息：
Journal of Graph Theory, Volume 90, Issue 4, 2019, Pages 547-560
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摘要：

令$\binom{X}{h}$为$n$ -集合$X\supseteq Y$的所有$h$ -子集的集合.给定一个集合$S\subseteq \binom{X}{h}$的着色（划分），我们感兴趣的是找到条件，在该条件下，该着色可扩展为$\binom{X}{h}$的着色，使得$X$的每个元素在每个颜色类（所有相同颜色的集合）中出现的次数与$r$相同。$S=\varnothing, r=1$的案例由西尔维斯特在18世纪进行研究，直到20世纪70年代仍然开放。案例$h=2,r=1$在文献中被广泛研究，并且与完成部分对称拉丁方密切相关。对于$S=\binom{Y}{h}$，我们完全解决了$h=4, |X|\geq 4.847323|Y|$和$h=5, |X|\geq 6.285214|Y|$的情况。此外，我们对解决$S=\binom{X}{h}\backslash \binom{Y}{h}$ .这些结果可以被看作是著名的Baranyai定理的推广，并朝着解决Cameron提出的一个40年前的问题取得了进展。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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