# On the Erd{\H o}s--Hajnal problem in the case of 3-graphs

## TL;DR

This paper investigates the minimal edges in 3-uniform hypergraphs that are not r-colorable, compares existing methods, and improves the lower bounds for this Erdős–Hajnal problem.

## Contribution

It provides a comparative analysis of methods and advances the lower bounds for the minimal edges in non-r-colorable 3-uniform hypergraphs.

## Key findings

- Comparison of existing methods for 3-uniform hypergraphs
- Improved lower bounds for m(3,r)
- Enhanced understanding of the Erdős–Hajnal problem in 3-graphs

## Abstract

Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. For the broad history of the problem see [RaiSh]. It is known that for a fixed $n$ the sequence \[ \frac{m(n,r)}{r^n} \] has a limit. The only trivial case is $n=2$ in which $m(2,r) = \binom{r+1}{2}$. In this note we focus on the case $n=3$. First, we compare the existing methods in this case and then improve the lower bound.