Coloring general Kneser graphs and hypergraphs via high-discrepancy hypergraphs

Abstract

We suggest a new method for coloring generalized Kneser graphs based on hypergraphs with high discrepancy and a small number of edges. The main result provides a proper coloring of $K(n,n∕2-t,s)$ in $(4+o\left(1\right)){(s+t)}^2$ colors, which is produced by Hadamard matrices. Also, we show that for colorings by independent set of a natural type, this result is the best possible up to a multiplicative constant.

Our method extends to Kneser hypergraphs as well.