In this talk, we discuss (weighted) sums of generalized polygonal numbers of the form
P_{a,m}(x):=sum a_j p_m(x_j),
where a_j are positive integers and x_j are integers, and p_m(x_j) is the x_j-th generalized m-gonal number. We are specifically interested in the classification of the vectors a and mgeq 3 for which the above sum is universal (it represents every positive integer with x running through all integer inputs). In recent years, certain "finiteness theorems" have been found for representations of integers by quadratic polynomials. From the famous Conway-Schneeberger 15 theorem, we know that for m=4 fixed, P_{a,4} is universal if and only if it represents every integer up to 15. For m=3 fixed, we have universality if and only if every integer up to 8 is represented, and for m=8 we know that universality holds if and only if every integer up to 60 is represented by a recent result of Ju and Oh. One can more generally show that a finiteness theorem exists for every m fixed. In other words, there exists a bound gamma_m such that P_{a,m} is universal if and only if it represents every integer up to gamma_m. The primary goal of this talk is to investigate the growth of gamma_m as a function of m. This is joint work with Jingbo Liu.