Tight universality of $m$-gonal forms with minimal criterion sets

For integers $m\ge 3$ and $n\ge 1$, an $m$-gonal form is called tight $\mathcal{T}(n)$-universal if it represents exactly the positive integers $\mathcal{T}(n)=\{n,n+1,n+2,\dots \}$. In this paper, we study the minimal criterion set $\mathrm{C}\mathrm{S}(m,n)$ for tight $\mathcal{T}(n)$-universality. Our main result determines $\mathrm{C}\mathrm{S}(m,n)$ for $n\ge 8$, $3\le m\le \lfloor \frac{3n+1}{2}\rfloor$, except for $(m,n)=(7,9)$ and $(7,10)$. More precisely, $$\mathrm{C}\mathrm{S}(m,n)=\{\begin{array}{ll}\{n,n+1,\dots ,2n-1\},& m=5,\\ \{n,n+1,\dots ,2n\},& m\mathrm{\ne }5.\end{array}$$ We also establish the corresponding tight $\mathcal{T}(n)$-universality results and show that the upper bound on $m$ is optimal.