Coloring general Kneser graphs and hypergraphs via high-discrepancy hypergraphs

TL;DR

This paper introduces a novel coloring method for generalized Kneser graphs using high-discrepancy hypergraphs, achieving near-optimal colorings with Hadamard matrices, and extends the approach to hypergraphs.

Contribution

The paper presents a new hypergraph-based coloring technique for Kneser graphs that improves existing bounds and extends to hypergraphs, utilizing Hadamard matrices.

Findings

Proper coloring of K(n, n/2-t, s) with (4 + o(1))(s + t)^2 colors

Colorings by independent sets are optimal up to a constant

Method extends to Kneser hypergraphs

Abstract

We suggest a new method on coloring generalized Kneser graphs based on hypergraphs with high discrepancy and small number of edges. The main result is providing a proper coloring of K(n, n/2-t, s) in (4 + o(1))(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.