Two questions on Kneser colorings

Abstract: In this paper, we investigate two questions on Kneser graphs $KG_{n,k}$. First, we prove that the union of $s$ intersecting families in ${[n]\choose k}$ has size at most ${n\choose k}-{n-s\choose k}$ 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 $\chi(KG_{n,k})=n-2k+2$. 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.