Two questions on Kneser colorings

Abstract

In this paper, we investigate two questions on Kneser graphs $K{G}_{n,k}$. First, we prove that the union of $s$ intersecting families in $\left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ has size at most $\left(\genfrac{}{}{0ex}{}{n}{k}\right)-\left(\genfrac{}{}{0ex}{}{n-s}{k}\right)$ for all sufficiently large $n$ that satisfy $n>(2+\epsilon )k^2+s$ with $\epsilon >0$. We provide an example that shows that this result is essentially tight for the number of colors close to $\chi (K{G}_{n,k})=n-2k+2$. We also improve the result of Bulankina and Kupavskii on the choice chromatic number, showing that it is at least $\frac{1}{25}n\log n$ for all $k<\sqrt{n}$ and $n$ sufficiently large.