2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59771In this paper we construct a family of simply connected, spin, non-complex, symplectic 4-manifolds which cover all but finitely many allowed lattice points $(χ, c)$ lying in $0 \leq c \leq 8.76χ$. Furthermore, as a corollary, we prove that there exist infinitely many exotic smooth structures on $(2n+1)(S^2 \times S^2)$ for all n large enough.This is a revised version. AMS-LaTeX file, 13 pages with 2 eps-figures. Minor errors and grammatical problems are corrected. To appear in Mathematische ZeitschriftGeometric Topology57R55, 57R57 (Primary) 57N13 (Secondary)text