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Partitioning all $k$-subsets into $r$-wise intersecting families
arXiv - CS - Discrete Mathematics Pub Date : 2021-07-27 , DOI: arxiv-2107.12741
Noga Alon

Let $r \geq 2$, $n$ and $k$ be integers satisfying $k \leq \frac{r-1}{r}n$. We conjecture that the family of all $k$-subsets of an $n$-set cannot be partitioned into fewer than $\lceil n-\frac{r}{r-1}(k-1) \rceil$ $r$-wise intersecting families. If true this is tight for all values of the parameters. The case $r=2$ is Kneser's conjecture, proved by Lov\'asz. Here we observe that the assertion also holds provided $r$ is either a prime number or a power of $2$.

中文翻译:

将所有 $k$-子集划分为 $r$-wise 相交族

令 $r \geq 2$, $n$ 和 $k$ 是满足 $k \leq \frac{r-1}{r}n$ 的整数。我们推测一个 $n$-set 的所有 $k$-子集的族不能被划分为少于 $\lceil n-\frac{r}{r-1}(k-1) \rceil$ $r $明智的相交家庭。如果为真,这对于参数的所有值都是严格的。情况 $r=2$ 是 Kneser 的猜想,由 Lov\'asz 证明。在这里我们观察到,如果 $r$ 是素数或 $2$ 的幂,则该断言也成立。
更新日期:2021-07-28
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