  In this paper, we classify the $(a_1,a_2,a_3,a_4,a_5)$ for which the
universality of an $m$-gonal form $F_m(\mathbf x)$ having its first five
coefficients as $(a_1,a_2,a_3,a_4,a_5)$ is characterized as the
representability of positive integers up to $m-4$ and see its applications.
