## Abstract

We say that a graph *F strongly arrows* a pair of graphs (*G,H*) and write *F*
*G,H*) if any 2-coloring 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 min{|*V* (*F*)| : *F*
*G,H*)}. We will consider two aspects of induced Ramsey numbers. Firstly we will show that the lower bound of the induced Ramsey number for a connected graph *G* with independence number α and a graph *H* with clique number *ω* is roughly *G* is not connected providing also a sharp lower bound which is linear in both parameters.