论文标题
Baranyai定理的不均匀扩展
A non-uniform extension of Baranyai's Theorem
论文作者
论文摘要
Baranyai的著名定理指出，当$ k $ dive $ n $时，所有$ k $ k $ k $ n $ k $ subsets的$ n $ emlement套件的$ k_n^k $可以分为完美的匹配。换句话说，$ k_n^k $是$ 1 $ -FACTORABLE。在本文中，我们确定所有$ n，k $，以便由$ k_n^{\ le k} $由$ [n] $的子集组成的$ k_n^{\ le k} $最高$ k $的子集组成$ 1 $ 1 $ -FACTOROR，因此将Baranyai定理扩展到了非统一设置。特别是，我们的结果意味着，对于固定的$ k $和足够大的$ n $，$ k_n^{\ le k} $是$ 1 $ - faltorable，当时仅当$ n \ equiv 0 $或$ -1 \ pmod k $。
A celebrated theorem of Baranyai states that when $k$ divides $n$, the family $K_n^k$ of all $k$-subsets of an $n$-element set can be partitioned into perfect matchings. In other words, $K_n^k$ is $1$-factorable. In this paper, we determine all $n, k$, such that the family $K_n^{\le k}$ consisting of subsets of $[n]$ of size up to $k$ is $1$-factorable, and thus extend Baranyai's Theorem to the non-uniform setting. In particular, our result implies that for fixed $k$ and sufficiently large $n$, $K_n^{\le k}$ is $1$-factorable if and only if $n \equiv 0$ or $-1 \pmod k$.
