Partitioning all $k$-subsets into $r$-wise intersecting families

Abstract

Let $r \geq 2$, $n$ and $k$ be integers satisfying $k \leq \frac{r-1}{r}n$. In the original arXiv version of this note we suggested a 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. We noted that if true this is tight for all values of the parameters, that the case $r=2$ is Kneser's conjecture, proved by Lov\'asz, and observed that the assertion also holds provided $r$ is either a prime number or a power of $2$. We have recently learned, however, that the assertion of the conjecture for all values of the parameters follows from a recent result of Azarpendar and Jafari \cite{AJ}.