Two questions on Kneser colorings

In this paper, we investigate two questions on Kneser graphs KGn,k. First, we prove that the union of s non-trivial intersecting families in [n] k has size at most n k-n-s k for all sufficiently large n that satisfy n>(2+ε)k² with ε>0. We provide an example that shows that this result is essentially tight for the number of colors close to χ(KGn,k)=n-2k+2. We also improve the result of Bulankina and Kupavskii on the choice chromatic number, showing that it is at least 116 nlog n for all k<√ n and n sufficiently large.