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In this paper, we investigate two questions on Kneser graphs KG₍, ₊. 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+) k² with >0. We provide an example that shows that this result is essentially tight for the number of colors close to (KG₍, ₊) =n-2k+2. We also improve the result of Bulankina and Kupavskii on the choice chromatic number, showing that it is at least 116 n n for all k< n and n sufficiently large.
Inozemtsev et al. (Tue,) studied this question.