On regular set systems containing regular subsystems,European Journal of Combinatorics

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On regular set systems containing regular subsystems
European Journal of Combinatorics ( IF 1 ) Pub Date : 2021-07-13 , DOI: 10.1016/j.ejc.2021.103393
Amin Bahmanian ₁ , Sadegheh Haghshenas ₂

Affiliation

Let $X, Y$ be finite sets, $r, s, h, λ \in N$ with $s \geq r, X ⊊ Y$ . By $λ (\binom{X}{h})$ we mean the collection of all $h$ -subsets of $X$ where each subset occurs $λ$ times. A coloring (partition) of $λ (\binom{X}{h})$ is $r$ -regular if each element of $X$ is in exactly $r$ subsets of each color. A one-regular color class is a perfect matching. We are interested in necessary and sufficient conditions under which an $r$ -regular coloring of $λ (\binom{X}{h})$ can be embedded into an $s$ -regular coloring of $λ (\binom{Y}{h})$ . Using algebraic techniques involving glueing together orbits of a suitably chosen cyclic group, the first author and Newman solved the case when $λ = 1, r = s, gcd (| X |, | Y |, h) = gcd (| Y |, h)$ . Using purely combinatorial techniques, we nearly settle the case $h = 4$ .

It is worth noting that completing partial symmetric latin squares is closely related to the case $λ = r = s = 1, h = 2$ which was solved by Cruse.

中文翻译：

在包含规则子系统的规则集系统上

让 $X, 是$ 是有限集， $r, 秒, H, λ \in N$ 和 $秒 \geq r, X ⊊ 是$ . 经过 $λ (\binom{X}{H})$ 我们的意思是所有的集合 $H$ -子集 $X$ 每个子集出现的地方 $λ$ 次。的着色（分区） $λ (\binom{X}{H})$ 是 $r$ -regular如果每个元素 $X$ 正是在 $r$ 每种颜色的子集。一个单一的常规颜色类是一个完美的匹配。我们感兴趣的必要和充分条件是 $r$ - 定期着色 $λ (\binom{X}{H})$ 可以嵌入到 $秒$ - 定期着色 $λ (\binom{是}{H})$ . Using algebraic techniques involving glueing together orbits of a suitably chosen cyclic group, the first author and Newman solved the case when $λ = 1, r = 秒, GCD (| X |, | 是 |, H) = GCD (| 是 |, H)$ . 使用纯粹的组合技术，我们几乎解决了这个案子 $H = 4$ .