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

TL;DR

This paper discusses a conjecture about partitioning all $k$-subsets of an $n$-set into $r$-wise intersecting families, connecting it to known results and recent proofs, and clarifying its validity across parameter ranges.

Contribution

It confirms that a conjecture on partitioning $k$-subsets into $r$-wise intersecting families follows from recent work, extending known cases to all parameters.

Findings

The conjecture is true for all parameter values due to recent results.

It generalizes the case $r=2$, known as Kneser's conjecture.

The assertion holds when $r$ is prime or a power of 2.

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}.