My apologies, I'm quite new to cryptography.
If we relax the definition of perfect secrecy such that for cyphertext $c$, messages $m_0$ and $m_1$, and constant $E$:
$P[c|m_0] \le E * P[c|m_1]$
Using this, Can I prove a lower bound on the size of the key space as a function of the size of the message space?
I don't understand how I can know anything about the size of the key space without further knowledge of the encryption procedure used and the size of messages and keys.