Colored Turán densities and finite directed templates

Abstract

Classical Turán theory relates the asymptotic extremal problem for an ordinary forbidden graph to its chromatic number. This principle fails when a forbidden copy must respect prescribed vertex colors, as occurs naturally in multipartite extremal problems and in embeddings with prescribed clusters. We study properly vertex-colored graphs whose hosts have a prescribed density between every pair of color classes, with containment interpreted through color-preserving homomorphisms. Our main theorem gives a finite directed reduction: for every fixed $m$ and $\varepsilon>0$, every sufficiently many-colored $d$-dense host admits a colored homomorphism from an $m$-layer half-blowup $D[m]$ of a finite digraph $D$ with density at least $d-\varepsilon$, where $D$ may be chosen with $\lceil\varepsilon^{-1}\rceil$ vertices. Consequently, the colored Turán density of every non-empty finite forbidden family equals the homomorphism Turán density of its associated acyclic orientations. We then determine the complete finite value set: these densities are exactly the Lagrangians of finite loopless digraphs, and hence are rational. The main technical input is a low-resolution sampling method that preserves all pairwise bipartite degree profiles on a large subfamily of color classes while keeping the sample size independent of the number of retained classes.