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 ε>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−ε, where D may be chosen with ⌈ε−1⌉ 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.
Related
- The ℓ-degree Turán densities of hypergraphs: infinite families and their finite subfamilies
- Improved Sidorenko bound for hypergraph Turan densities
- Vertex-colored Turán theorems with applications in extremal hypergraph problems
- Weighted Turán densities for hypergraphs
- On the structure of uniform Turán densities
BibTeX
@misc{chen2026colored,
title = {{Colored Turán densities and finite directed templates}},
author = {Nannan Chen and Ander Lamaison and Heng Li and Hong Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17992},
eprint = {2610.17992},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17992},
}
RIS
TY - UNPB
TI - Colored Turán densities and finite directed templates
AU - Nannan Chen
AU - Ander Lamaison
AU - Heng Li
AU - Hong Liu
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17992
UR - https://arxiv.org/abs/2610.17992
AB - 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 ε>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−ε, where D may be chosen with ⌈ε^−1⌉ 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.
ER -