The chromatic number of 3-stable Kneser graphs

For an integer $s \ge 2$, a subset $S \subseteq [n]$ is {\em $s$-stable} if $\min \{j - i, n + i - j\}\ge s$ for every $i,j \in S$ with $i<j$.

Denote the set of all $s$-stable subsets of size $k$ of $[n]$ by $\binom{[n]}{k}_{s\text{-stable}}$.

Schrijver proved in 1978 that whenever $n\ge 2k$, the chromatic number of the Kneser graph $\mathrm{KG}\big( \binom{[n]}{k}_{2\text{-stable}}\big)$ is $n - 2k +2$.

Generalizing this result, Meunier conjectured in 2011 that $\chi\left( \mathrm{KG}\big( \binom{[n]}{k}_{s\text{-stable}} \big) \right)= n - sk +s$ for all $n\ge sk$.

This conjecture was previously proven for all even $s$, for $s \ge 4$ and large enough $n$, and for $k=2$.

We prove the conjecture in the cases $s=3$ and $n$ large enough, or $k=s=3$.

To this end, we prove versions of the Hilton-Milner theorem for $s$-stable sets.

We also present a topological approach towards Meunier's conjecture.