  A graph is a $k$-threshold graph with thresholds $\theta_1, \theta_2, \dots,
\theta_k$ if we can assign a real number $r_v$ to each vertex $v$ such that for
any two distinct vertices $u$ and $v$, $uv$ is an edge if and only if the
number of thresholds not exceeding $r_u+r_v$ is odd. The threshold number of a
graph is the smallest $k$ for which it is a $k$-threshold graph. Multithreshold
graphs were introduced by Jamison and Sprague as a generalization of classical
threshold graphs. They asked for the exact threshold numbers of complete
multipartite graphs. Recently, Chen and Hao solved the problem for complete
multipartite graphs where each part is not too small, and they asked for the
case when each part has size $3$. We determine the exact threshold numbers of
$K_{3, 3, \dots, 3}$, $K_{4, 4, \dots, 4}$ and their complements $nK_3$,
$nK_4$. This improves a result of Puleo.
