论文标题
在包含常规子系统的常规设置系统上
On Regular Set Systems Containing Regular Subsystems
论文作者
论文摘要
令$ x，y $为有限套件，$ r，s，h，λ\ in \ mathbb {n} $，带有$ s \ geq r，x \ subsetneq y $。由$λ\ binom {x} {h} $，我们的意思是收集$ x $的所有$ h $ -subsets，其中每个子集发生$λ$ times。 $λ\ binom {x} {h} $的着色是{\ it $ r $ - regular}，如果在每个颜色类中，$ x $的每个元素都会发生$ r $ $ times。一个规范的颜色类是{\ it完美匹配}。我们对$ r $ r $的$λ\ binom {x} {h} $的$ r $定型着色可以嵌入到$ s $ s $ - $λ\ binom {y} {h} $的$ s $ regular contreding中。使用代数技术涉及将适当选择的环状组胶合在一起的轨道，第一作者和纽曼（Combinatorica 38（2018），第6、1309---1335号），当$λ= 1，r = s，r = s，r = s，\ gcd（| x | x |，| y | y | y | y | y | y | y | y | y | y | y | y |，使用纯粹的组合技术，我们几乎解决了$ h = 4 $的情况。两个主要的挑战包括找到所有必要的条件，并获得$ | y | $的确切约束。 值得注意的是，完成部分对称拉丁正方形与$λ= r = s = 1，h = 2 $密切相关，这是由Cruse解决的（J. comb。理论Ser。A16（1974），18--22）。
Let $X,Y$ be finite sets, $r,s,h, λ\in \mathbb{N}$ with $s\geq r, X\subsetneq Y$. By $λ\binom{X}{h}$ we mean the collection of all $h$-subsets of $X$ where each subset occurs $λ$ times. A coloring of $λ\binom{X}{h}$ is {\it $r$-regular} if in every color class each element of $X$ occurs $r$ times. A one-regular color class is a {\it perfect matching}. We are interested in the 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 (Combinatorica 38 (2018), no. 6, 1309--1335) solved the case when $λ=1,r=s, \gcd (|X|,|Y|,h)=\gcd(|Y|,h)$. Using purely combinatorial techniques, we nearly settle the case $h=4$. Two major challenges include finding all the necessary conditions, and obtaining the exact bound for $|Y|$. 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 (J. Comb. Theory Ser. A 16 (1974), 18--22).
