# Generalized Ramsey numbers: forbidding paths with few colors

## TL;DR

This paper studies a generalized Ramsey number function that measures the minimum colors needed in edge-colorings of complete graphs to ensure paths have a minimum number of colors, extending previous work to new graph types.

## Contribution

It determines the asymptotic behavior of the function for paths on vertices for most values of path length and color count, broadening understanding beyond complete graphs.

## Key findings

- Established asymptotics for f(K_n, P_v, q) for most v and q
- Extended extremal function analysis to paths beyond complete graphs
- Provided new bounds and exact values for specific cases

## Abstract

Let $f(K_n, H, q)$ be the minimum number of colors needed to edge-color $K_n$ so that every copy of $H$ is colored with at least $q$ colors. Originally posed by Erd\H{o}s and Shelah when $H$ is complete, the asymptotics of this extremal function have been extensively studied when $H$ is a complete graph or a complete balanced bipartite graph. Here we investigate this function for some other $H$, and in particular we determine the asymptotic behavior of $f(K_n, P_v, q)$ for almost all values of $v$ and $q$, where $P_v$ is a path on $v$ vertices.