Trivial colors in colorings of Kneser graphs

TL;DR

This paper proves that for large enough parameters, any proper coloring of Kneser graphs with a near-minimal number of colors must include a trivial color, advancing understanding of their coloring structure.

Contribution

It establishes a near-tight bound on when trivial colors must appear in proper colorings of Kneser graphs, generalizing previous results.

Findings

Any proper coloring with n-2k+2 colors contains a trivial color for n > (2+ε)k^2

The bound on n is essentially tight

Results extend to the minimum number of non-trivial colors needed

Abstract

We show that any proper coloring of a Kneser graph $KG_{n,k}$ with $n-2k+2$ colors contains a trivial color (i.e., a color consisting of sets that all contain a fixed element), provided $n>(2+\varepsilon)k^2$, where $\varepsilon\to 0$ as $k\to \infty$. This bound is essentially tight. This is a consequence of a more general result on the minimum number of non-trivial colors needed to properly color $KG_{n,k}$.