A Note on Lower Bounds for Induced Ramsey Numbers

TL;DR

This paper investigates lower bounds for induced Ramsey numbers, establishing sharp bounds based on graph parameters like independence and clique numbers, with implications for both connected and disconnected graphs.

Contribution

It provides new sharp lower bounds for induced Ramsey numbers involving parameters like independence and clique numbers, extending to disconnected graphs.

Findings

Lower bound for connected graphs: approximately (ω²α)/2

Bounds are sharp for connected graphs

Linear lower bounds for disconnected graphs

Abstract

We say that a graph $F$ strongly arrows a pair of graphs $(G,H)$ if any 2-colouring of its edges with red and blue leads to either a red $G$ or a blue $H$ appearing as induced subgraphs of $F$. The induced Ramsey number, $IR(G,H)$ is defined as the minimum number of vertices of a graph $F$ which strongly arrows a pair $(G,H)$. We will consider two aspects of induced Ramsey numbers. Firstly there will be shown that the lower bound of the induced Ramsey number for a connected graph $G$ with independence number $\alpha$ and a graph $H$ with clique number $\omega$ roughly $\frac{\omega^2\alpha}{2}$. This bounds is sharp. Moreover we discuss also the case when $G$ is not connected providing also a sharp lower bound which is linear in both parameters