A natural generalisation in graph Ramsey theory

TL;DR

This paper explores a generalization in graph Ramsey theory, analyzing graphs with specific colouring properties, establishing construction methods, and characterizing minimal graphs related to cycles, advancing understanding of Ramsey-infiniteness.

Contribution

It introduces a new class of graphs with colouring properties, proves methods to construct these graphs, and characterizes minimal graphs when considering cycle properties.

Findings

Established a method to build G_r from G_{r+1} for H=K_n.

Proved the Ramsey-infiniteness of these graphs.

Characterized minimal G_r graphs when H contains a cycle.

Abstract

In this note we study graphs $G_r$ with the property that every colouring of $E(G_r)$ with $r+1$ colours admits a copy of some graph $H$ using at most $r$ colours. For $1\le r\le e(H)$ such graphs occur naturally at intermediate steps in the synthesis of a $2$-colour Ramsey graph $G_1\longrightarrow H$. (The corresponding notion of Ramsey-type numbers was introduced by Erd\"os, Hajnal and Rado in 1965 and subsequently studied by Erd\"os and Szemer\'edi in 1972). For $H=K_n$ we prove a result on building a $G_{r}$ from a $G_{r+1}$ and establish Ramsey-infiniteness. From the structural point of view, we characterise the class of the minimal $G_r$ in the case when $H$ is relaxed to be the graph property of containing a cycle; we then use it to progress towards a constructive description of that class by proving both a reduction and an extension theorem.