Monochromatic tree covers and Ramsey numbers for set-coloured graphs

TL;DR

This paper extends classical Ramsey theory to graphs with edges coloured by sets of colours, providing bounds for monochromatic tree covers and generalising Ramsey numbers, with implications for hypergraph transversals.

Contribution

It introduces bounds for monochromatic tree covers in set-coloured graphs and generalises Ramsey numbers, connecting these to hypergraph transversal problems.

Findings

Bounds for monochromatic tree covers in complete graphs and bipartite graphs.

A stronger version of Ryser's conjecture for certain hypergraphs.

The bound r-k is not optimal in all cases.

Abstract

We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a {\em set} of colours (instead of just one colour). We give bounds for monochromatic tree covers in this setting, both for an underlying complete graph, and an underlying complete bipartite graph. We also discuss a generalisation of Ramsey numbers to our setting and propose some other new directions. Our results for tree covers in complete graphs imply that a stronger version of Ryser's conjecture holds for $k$-intersecting $r$-partite $r$-uniform hypergraphs: they have a transversal of size at most $r-k$. (Similar results have been obtained by Kir\'aly et al., see below.) However, we also show that the bound $r-k$ is not best possible in general.