A note on hardness of promise hypergraph colouring

TL;DR

This paper simplifies the proof of NP-hardness for coloring certain hypergraphs, extending the algebraic framework for Promise CSPs with a weaker PCP theorem.

Contribution

It provides a simpler proof of a known NP-hardness result for hypergraph coloring within the Promise CSPs framework.

Findings

NP-hardness of c-coloring 2-colorable 3-uniform hypergraphs for all c ≥ 2

Recasts existing proof in algebraic Promise CSPs framework

Uses a weaker version of the PCP theorem

Abstract

We show a slightly simpler proof the following theorem by I. Dinur, O. Regev, and C. Smyth: for all $c \geq 2$, it is NP-hard to find a $c$-colouring of a 2-coloruable 3-uniform hypergraph. We recast this result in the algebraic framework for Promise CSPs, using only a weaker version of the PCP theorem.