Beyond the broken tetrahedron

TL;DR

The work addresses the uniform Turán problem for 3-uniform hypergraphs by formulating embeddings within reduced hypergraphs. It introduces a novel row-preparation and vertex-gluing framework to force reduced images of target hypergraphs in uniformly dense reduced hypergraphs, yielding exact density results for $F_{\star}$. Specifically, it proves $\pi_{x}(F_{\star})=\tfrac{1}{4}$ and demonstrates a first gluing step (one-vertex glue) for densities above $\tfrac{1}{4}$, outlining a path toward determining $\pi_{x}(K_4^{(3)})$. The methods—rooted in reduced hypergraph analysis, cleaning arguments, and Ramsey-type arguments—advance the toolkit for uniform-density hypergraph Turán problems and illuminate how intermediate reduced-hypergraph steps can bridge toward the open target of $K_4^{(3)}$.

Abstract

Here we consider the hypergraph Turán problem in uniformly dense hypergraphs as was suggested by Erdős and Sós. Given a $3$-graph $F$, the uniform Turán density $π_u(F)$ of $F$ is defined as the supremum over all $d\in[0,1]$ for which there is an $F$-free uniformly $d$-dense $3$-graph, where uniformly $d$-dense means that every linearly sized subhypergraph has density at least $d$. Recently, Glebov, Král', and Volec and, independently, Reiher, Rödl, and Schacht proved that $π_u(K_4^{(3)-})=\frac{1}{4}$, solving a conjecture by Erdős and Sós. Despite substantial attention, the uniform Turán density is still only known for very few hypergraphs. In particular, the problem due to Erdős and Sós to determine $π_u(K_4^{(3)})$ remains wide open. In this work, we determine the uniform Turán density of the $3$-graph on five vertices that is obtained from $K_4^{(3)-}$ by adding an additional vertex whose link forms a matching on the vertices of $K_4^{(3)-}$. Further, we point to two natural intermediate problems on the way to determining $π_u(K_4^{(3)})$, and solve the first of these.

Figures (5)

• Figure 1.1: The hypergraph $F_{\star}$.
• Figure 1.2: The reduced image of $K_4^{(3)-}$ with an additional edge and one vertex 'glued' ($\alpha_{34})$.
• Figure 3.1: $F_{\star}$ embedded across several prepared rows in the reduced hypergraph $\mathcal{A}$. Blue lines represent edges in $Q^r$.
• Figure 3.2: The reduced image of $F_{\star}$ in the reduced hypergraph $\mathcal{A}$. As before, the gray lines between two indices $i$ and $j$ represent the vertex class $\mathcal{P}^{ij}$.
• Figure 4.1: $K_4^{(3)-}$ with one vertex glued embedded in the reduced hypergraph $\mathcal{A}$ that we obtain, following the procedure above.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• proof : Proof of Theorem \ref{['theorem:main']}
• Lemma 3.1
• proof
• Lemma 3.2