A lower bound on the size of an absorbing set in an arc-coloured tournament

TL;DR

This paper establishes a lower bound on the size of an absorbing set in an arc-coloured tournament, advancing understanding of monochromatic path structures in such graphs.

Contribution

It provides the first known lower bound on the minimal size of an absorbing set in arc-coloured tournaments, complementing existing upper bounds.

Findings

Established a lower bound on f(n) for arc-coloured tournaments.

Demonstrated the existence of large absorbing sets in certain colourings.

Enhanced theoretical understanding of monochromatic path coverage.

Abstract

Bousquet, Lochet and Thomass\'e recently gave an elegant proof that for any integer $n$, there is a least integer $f(n)$ such that any tournament whose arcs are coloured with $n$ colours contains a subset of vertices $S$ of size $f(n)$ with the property that any vertex not in $S$ admits a monochromatic path to some vertex of $S$. In this note we provide a lower bound on the value $f(n)$.