Two questions on Kneser colorings

TL;DR

This paper explores bounds on the size of intersecting families in Kneser graphs and improves lower bounds on their choice chromatic number, advancing understanding of their combinatorial properties.

Contribution

It proves a tight upper bound on the union size of intersecting families in Kneser graphs and improves the lower bound on the choice chromatic number for large n.

Findings

Union size of s intersecting families is at most ${nrace k}-{n-srace k}$ for large n.

Choice chromatic number is at least $rac{1}{25} n \log n$ for $k<\sqrt n$ and large n.

Results are tight for color counts near the chromatic number.

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.