On the number of $p'$-degree characters in a finite group
  Let $p$ be a prime divisor of the order of a finite group $G$. Then $G$ has at least $2 \sqrt{p-1}$ complex irreducible characters of degrees prime to $p$. In case $p$ is a prime with $\sqrt{p-1}$ an integer this bound is sharp for infinitely many groups $G$. 
Gunter Malle and Attila Mar\'oti
math.GR math.RT