For every 3-uniform hypergraph F, pi_co(F)=0 if and only if F is layered and pi(F)=0, where pi(F) is the uniform Turán density and pi_co(F)=supgamma∈[0,1):for arbitrarily large n there is an n-vertex F-free 3-graph H with delta_2(H)≥gamma n. Here F is layered if its vertices can be labelled so that every edge has a unique maximum-labelled vertex and any two edges with the same maximum label have the same multiset of vertex labels.

Ding–Lamaison–Liu–Wang–Yang layered-graph conjecture

For every $3$ -uniform hypergraph $F$ , $\pi_{\mathrm{co}}(F)=0$ if and only if $F$ is layered and $\pi(F)=0$ , where $\pi(F)$ is the uniform Turán density and $\pi_{\mathrm{co}}(F)=\sup\{\gamma\in[0,1):\text{for arbitrarily large }n\text{ there is an }n\text{-vertex }F\text{-free }3\text{-graph }H\text{ with }\delta_2(H)\geq\gamma n\}$ . Here $F$ is layered if its vertices can be labelled so that every edge has a unique maximum-labelled vertex and any two edges with the same maximum label have the same multiset of vertex labels.

Progress summary

Solved

The original three-uniform version is now proved, while the broader all-rank statement has only its necessary direction.

The conjecture asks whether zero codegree Turán density is characterized by a layered structure. Ding, Lamaison, Liu, Wang, and Yang posed the relevant question for $3$ -graphs; recent work settles that formulation.

Known results

Ding, Lamaison, Liu, Wang, and Yang established vanishing codegree-density results for a class of layered $3$ -graphs; the retrieved source gives no year.

August 2026 proof of the $3$ -graph equivalence

A new arXiv paper proves that every non-layered $k$ -graph has positive codegree Turán density, with an explicit lower bound, while constructing examples with arbitrarily small positive density. Consequently, the conjectured equivalence follows for $k=3$ .

Current status (as of August 2026): The conjectured characterization is resolved for $3$ -graphs; for general $k$ , the result establishes only that zero density forces layering, so the converse remains open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Any k-graph with zero ℓ-degree Turán density is layered — arXiv

extremal-combinatorics hypergraph-theory

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