In this paper, we investigate two questions on Kneser graphs $KG_{n,k}$. First, we prove that the union of $s$ non-trivial intersecting families in ${[n]\choose k}$ has size at most ${n\choose k}-{n-s\choose k}$ for all sufficiently large $n$ that satisfy $n>(2+\epsilon)k^2$ 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 $\frac 1{16} n\log n$ for all $k<\sqrt n$ and $n$ sufficiently large.

翻译：本文研究了克内泽尔图 $KG_{n,k}$ 上的两个问题。首先，我们证明：对于所有足够大的 $n$（满足 $n>(2+\epsilon)k^2$，其中 $\epsilon>0$），在 ${[n]\choose k}$ 中，$s$ 个非平凡相交族的并集的大小最多为 ${n\choose k}-{n-s\choose k}$。我们通过一个例子表明，当颜色数接近 $\chi(KG_{n,k})=n-2k+2$ 时，这一结果本质上是紧的。此外，我们改进了布兰金娜和库帕夫斯基关于选择色数的结果，证明对于所有 $k<\sqrt n$ 且 $n$ 足够大时，选择色数至少为 $\frac 1{16} n\log n$。