Universal Sums of m-Gonal Numbers

In this paper we study universal quadratic polynomials, which arise as sums of polygonal numbers. Specifically, we determine an asymptotic upper bound (as a function of
$m$
) on the size of the set
$S_m\subset{\mathbb{N}}$
such that if a sum of
$m$
-gonal numbers represents
$S_m$
, then it represents
${\mathbb{N}}$
.