# An explicit edge-coloring of K_n with six colors on every K_5

## TL;DR

This paper presents a new explicit edge-coloring method for complete graphs that reduces the number of colors needed to prevent small monochromatic cliques, improving previous bounds significantly.

## Contribution

The authors provide an explicit construction for coloring K_n with fewer colors to avoid certain monochromatic subgraphs, surpassing prior probabilistic bounds.

## Key findings

- f(n,5,6) < n^(1/2+o(1))
- Improves upon previous O(n^(3/5)) bound
- Establishes a lower bound of Ω(n^(1/2))

## Abstract

For fixed integers p and q, let f(n,p,q) denote the minimum number of colors needed to color all of the edges of the complete graph K_n such that no clique of p vertices spans fewer than q distinct colors. A construction is given which shows that f(n,5,6) < n^(1/2+o(1)). This improves upon the best known probabilistic upper bound of O(n^(3/5)) given by Erd\H{o}s and Gy\'arf\'as. It is also shown that f(n,5,6) = {\Omega}(n^(1/2)).