For $m=3,4,\dots$, let $P_m(x)=((m-2)x^2-(m-4)x)/2$. A quadratic polynomial of the form $$ f=f(x,y,z)=aP_m(x)+bP_m(y)+cP_m(z)\quad (a,b,c\in \mathbb{N}) $$ is called a ternary sum of generalized $m$-gonal numbers. An integer $n$ is said to be represented by $f$ if the Diophantine equation $$ f(x,y,z)=n $$ has an integer solution. We say that $f$ is {\it regular} if the local-global principle holds for the representations of all nonnegative integers by $f$. In this talk, we establish an explicit upper bound for $m$ such that there exists a regular ternary sum of generalized $m$-gonal numbers.