# Simplified inpproximability of hypergraph coloring via t-agreeing families

## TL;DR

This paper presents new simplified proofs of the hardness of hypergraph coloring problems, leveraging bounds on extremal t-agreeing families, and establishes quasi NP-hardness for various coloring scenarios.

## Contribution

It introduces a unified technique based on t-agreeing family bounds to reprove hypergraph coloring hardness results, simplifying previous proofs and extending the range of hardness results.

## Key findings

- Proves quasi NP-hardness for coloring 3-colorable 4-uniform hypergraphs with polylogarithmic colors.
- Establishes hardness for coloring 3-colorable 3-uniform hypergraphs with sub-logarithmic colors.
- Shows quasi NP-hardness for coloring 2-colorable 6-uniform hypergraphs with polylogarithmic colors.

## Abstract

We reprove the results on the hardness of approximating hypergraph coloring using a different technique based on bounds on the size of extremal $t$-agreeing families of $[q]^n$. Specifically, using theorems of Frankl-Tokushige [FT99], Ahlswede-Khachatrian [AK98] and Frankl [F76] on the size of such families, we give simple and unified proofs of quasi NP-hardness of the following problems: $\bullet$ coloring a $3$ colorable $4$-uniform hypergraph with $(\log n)^\delta$ many colors $\bullet$ coloring a $3$ colorable $3$-uniform hypergraph with $\tilde{O}(\sqrt{\log \log n})$ many colors $\bullet$ coloring a $2$ colorable $6$-uniform hypergraph with $(\log n)^\delta$ many colors $\bullet$ coloring a $2$ colorable $4$-uniform hypergraph with $\tilde{O}(\sqrt{\log \log n})$ many colors where $n$ is the number of vertices of the hypergraph and $\delta>0$ is a universal constant.