  For primes $p$, we study the maximal possible size of a $p$-core
$p'$-partition (a partition with no hook lengths or parts divisible by $p$).
McDowell recently proved that the maximum is attained by a unique partition,
say $\Lambda_p$. Using his graph theoretic description of $\Lambda_p$, we prove
for $p > 10^6$ that \[\frac{1}{24}p^6 - p^5\sqrt{p} < |\Lambda_p| <
\frac{1}{24}p^6 - \frac{1}{200}p^5\sqrt{p},\] which shows that $|\Lambda_p|
\sim p^6/24$ as $p \to \infty$.
