# King-serf duo by monochromatic paths in k-edge-coloured tournaments

## TL;DR

This paper investigates monochromatic paths in edge-colored tournaments, establishing bounds on vertex sets that ensure short monochromatic paths to or from these sets, extending classical conjectures in combinatorics.

## Contribution

It proves the existence of small vertex sets in any k-edge-colored tournament that guarantee short monochromatic paths to or from these sets, generalizing Erdős's conjecture.

## Key findings

- Existence of vertex sets with bounded total size for monochromatic paths
- Applicable to both finite and infinite cardinals
- Extends classical conjectures in monochromatic path theory

## Abstract

An open conjecture of Erd\H{o}s states that for every positive integer $k$ there is a (least) positive integer $f(k)$ so that whenever a tournament has its edges colored with $k$ colors, there exists a set $S$ of at most $f(k)$ vertices so that every vertex has a monochromatic path to some point in $S$. We consider a related question and show that for every (finite or infinite) cardinal $\kappa>0$ there is a cardinal $ \lambda_\kappa $ such that in every $\kappa$-edge-coloured tournament there exist disjoint vertex sets $K,S$ with total size at most $ \lambda_\kappa$ so that every vertex $ v $ has a monochromatic path of length at most two from $K$ to $v$ or from $v$ to $S$.