  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
