  We prove that the number of irreducible ordinary characters in the principal
$p$-block of a finite group $G$ of order divisible by $p$ is always at least
$2\sqrt{p-1}$. This confirms a conjecture of H\'{e}thelyi and K\"{u}lshammer
for principal blocks and provides an affirmative answer to Brauer's Problem 21
for principal blocks of bounded defect. Our proof relies on recent works of
Mar\'{o}ti and Malle-Mar\'{o}ti on bounding the conjugacy class number and the
number of $p'$-degree irreducible characters of finite groups, earlier works of
Brou\'{e}-Malle-Michel and Cabanes-Enguehard on the distribution of characters
into unipotent blocks and $e$-Harish-Chandra series of finite reductive groups,
and known cases of the Alperin-McKay conjecture.
