Scaling properties of maximally compact chains

Scaling of the exact function for the number of intramolecular nonbonded contacts in a single maximally compact linear homopolymer on hypercubic lattices is determined as a function of number N of monomers and dimension d. A representative maximally compact structure is designed and an exact recursive expression for the maximum number $m_{\max }$ of contacts is derived from that design. The equivalent nonrecursive expression yields the asymptotic scaling of $m_{\max }$ as $\left(d-1\right)N-{\mathrm{dN}}^{\Delta }+1,$ with $\Delta =\left(d-1\right)/d.$ Implications in polymer and protein studies are discussed.