Uniform Tur\'an densities of $k$-uniform hypergraphs

TL;DR

This paper introduces a palette framework to compute and classify uniform Turán densities of hypergraphs, providing exact values and examples of non-principal families without relying on hypergraph regularity.

Contribution

It develops a new palette homomorphism framework for Turán density calculations, simplifying the process and enabling new results for hypergraph families.

Findings

Established exact Turán densities for specific hypergraph families.

Developed palette classification tools for hypergraph colorability constraints.

Provided the first examples of non-principal families with Turán densities.

Abstract

For $k\ge 3$, the $(k-2)$-uniform Tur\'an density $\pi_{k-2}(F)$ of a $k$-graph $F$ is the supremum of $d$ for which there are arbitrarily large $F$-free $k$-graphs that are uniformly $d$-dense with respect to the $k$-vertex cliques of every $(k-2)$-graph on the same vertex set. We develop a \emph{palette framework} for this density. For every family $\mathcal F$ of $k$-graphs, we prove that $\pi_{k-2}(\mathcal F)$ equals the corresponding palette Tur\'an density. We further establish palette classification tools for the existence of $k$-graphs satisfying prescribed palette colorability constraints. Those together allow us to reduce exact density computations to a palette-homomorphism framework without relying on the hypergraph regularity method. As applications, for all $k\ge 3$ and $r\ge 2$, we establish the following values \[ \frac{r-1}{r},\quad \frac{(r-1)^2}{r^2},\quad \frac{r-1}{2r},\quad \frac{(k-1)^k}{k^k},\quad \frac{4(k-2)^{k-2}}{k^k},\quad \frac{4(k-2)^{k-2}}{3k^k} \] as $(k-2)$-uniform Tur\'an densities of single $k$-graphs. Finally, for every $k\ge3$, we show that there exist $k$-graphs $F_1,F_2$ such that \[ \pi_{k-2}(\{F_1,F_2\})< \min\{\pi_{k-2}(F_1),\pi_{k-2}(F_2)\}, \] which provides the first examples of \emph{non-principal} families for this density.