Schacht’s and Lamaison’s uniform Turán-density questions

For every $3$ -uniform hypergraph $F$ , determine when its cherry Turán density vanishes. The claimed characterization is $\pi_{ee}(F)=0$ if and only if $F$ is max-layered. The corresponding zero-jump questions ask whether there is a constant $c>0$ such that no $3$ -uniform hypergraph $F$ satisfies $\pi_{\star}(F)\in(0,c)$ . The claimed answers are that zero is a jump for $\pi_{vvv}$ , but not for $\pi_{ve}$ or $\pi_{ee}$ ; equivalently, for every $c>0$ there exists a $3$ -uniform hypergraph $F$ with $0<\pi_{ve}(F)<c$ , and likewise one with $0<\pi_{ee}(F)<c$ .

Equivalent formulations

For every $3$ -uniform hypergraph $F$ , the following conditions are equivalent: $\pi_{ee}(F)=0$ , $\pi_{ee}^{\rm pal}(F)=0$ , and $F$ is max-layered.
source: Vanishing Uniform Turán Densities: Characterization and Jumps

References

arXiv

Vanishing Uniform Turán Densities: Characterization and Jumps — arXiv — Hao Lin, Jie Ma, Wenling Zhou

extremal-combinatorics hypergraph-theory

Posted 3 hours ago · edited 3 hours ago

Progress summary

Refreshed 3 hours ago

Open · solution reported

A new unpublished paper claims to settle the main structural and jump questions, but its mathematics has not been independently checked.

The questions concern when uniform Turán densities vanish and how they behave near zero. A 2026 preprint claims a general characterization that includes the max-layered criterion and resolves the associated palette- and jump-comparison questions.

Known results

Reiher, Rödl, and Schacht (2018) characterized $3$ -graphs with $b[?25l\pi_1(F)=0$ and showed that failure of the condition forces $b[?25l\pi_1(F)\ge 1/27$ .

2026 preprint: claimed structural resolution

The preprint claims, for every $k\ge 3$ , $0\le t\le k-1$ , and every $k$ -graph $F$ , that $b[?25l\pi_t^{\mathrm{ext}}(F)=0$ exactly when $F$ is $t$ -layered. It therefore claims that, for $3$ -graphs, vanishing cherry density is equivalent to being max-layered, and reports corresponding palette-density and zero-jump results. The claim is unrefereed and lacks independent mathematical assessment; the authors say unnamed AI tools generated the general-case argument, which they independently checked.

Current status (as of October 2026): The structural and jump questions are claimed solved by an unrefereed preprint, but the claimed resolution remains unverified.

Sources

No solutions have been posted yet.