Any $k$-graph with zero $\ell$-degree Turán density is layered

Abstract: The codegree Turán density $π<em>{\mathrm{co}}(F)$ is the supremum over all $γ\in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $γn$ edges. Ding, Lamaison, Liu, Wang, and Yang (JLMS, 2025) studied the problem of what 3-graphs $F$ satisfy $π</em>{\mathrm{co}}(F) = 0$. They introduced layered $3$-graphs and conjectured that a $3$-graph has zero codegree Turán density if and only if it is layered and has zero uniform Turán density. For $k\ge 3$, a $k$-graph is called layered if its vertices can be labelled so that every edge has a unique maximum label and two edges with the same maximum label have the same label multiset. In this paper, we show that every non-layered $k$-graph $F$ on $m$ vertices satisfies [ π{\mathrm{co}}(F)\ge q{k,m}^{-q_{k,m}}>0, \quad \text{where}\quad q_{k,m}=\frac{(k-1)^{m+1}-1}{k-2}, ] which implies any $k$-graph with zero $\ell$-degree Turán density is layered, and the case $k=3$ confirms the conjecture of Ding, Lamaison, Liu, Wang, and Yang.

• The paper proves that every non-layered k-graph F with m vertices has positive codegree Turán density, with the explicit lower bound π_co(F) ≥ q_{k,m}^{−q_{k,m}}, where q_{k,m}=((k−1)^{m+1}−1)/(k−2).
• The authors translate layeredness into quotient-digraph consistency and use admissibly labelled complete (k−1)-ary trees to construct F-free host k-graphs with linear minimum codegree.
• The result shows that zero ℓ-degree Turán density always forces layeredness for every uniformity and degree parameter, while leaving the sharp quantitative bounds and higher-uniformity characterizations open.

Background and context

The codegree Turán density $\pi_{\mathrm{co}}(F)$, introduced by Mubayi and Zhao [mubayi], measures how large a uniform minimum codegree an $F$-free $k$-graph can sustain. Determining exact values is notoriously difficult: $\pi_{\mathrm{co}}(K_4^{(3)})$ remains open despite decades of work, and only a handful of exact values are known (e.g., $\pi_{\mathrm{co}}(\mathbb{F})=1/2$ for the Fano plane [Mubayi2005] and $\pi_{\mathrm{co}}(F_{3,2})=1/3$ [FRMPV2015]). A more tractable question, posed by Ding, Lamaison, Liu, Wang, and Yang [DLLWY], is which 3-graphs have vanishing codegree Turán density. They introduced layered 3-graphs—those admitting a vertex labelling in which every edge has a unique maximum label and edges sharing a maximum label share the same label multiset—and conjectured that $\pi_{\mathrm{co}}(F)=0$ if and only if $F$ is layered with zero uniform Turán density $\pi(F)$.

This paper by Yang, Fang, and Chen resolves that conjecture affirmatively, and does so in a stronger form valid for all uniformities and all $\ell$-degree Turán densities simultaneously.

Main result

The central theorem states that for $F$0, any non-layered $F$1-graph $F$2 on $F$3 vertices satisfies

$F$4

Two consequences follow immediately. First, since $F$5 and $F$6 for all $F$7, every $F$8-graph with zero $F$9-degree Turán density must be layered—a structural dichotomy that holds uniformly across the entire degree spectrum. Second, combined with the earlier results of Ding et al. (that layeredness plus vanishing uniform density implies vanishing codegree density, and that vanishing codegree density implies vanishing uniform density), the case $k$0 confirms both of their conjectures: a 3-graph has $k$1 exactly when it is layered with $k$2, and a linear 3-graph has $k$3 exactly when it is layered (using Reiher–Rödl–Schacht's characterization that linear 3-graphs have vanishing uniform density).

It should be noted that the quantitative bound is extremely weak: $k$4 decays super-exponentially in $k$5. The theorem's value is qualitative—it certifies positivity—but says essentially nothing about the actual magnitude of $k$6 for specific non-layered graphs.

Proof architecture

The proof proceeds through three conceptual stages.

Layeredness as quotient-digraph consistency. The authors first recast layeredness combinatorially. Given a $k$7-to-$k$8 orientation of a $k$9-graph (each edge oriented toward one designated head vertex) and a partition $\pi_{\mathrm{co}}(K_4^{(3)})$0 of the vertices, they form a loopless quotient digraph whose arcs record tail-class-to-head-class incidences. Lemma 2 establishes that $\pi_{\mathrm{co}}(K_4^{(3)})$1 is layered if and only if there exist such an orientation and partition satisfying two conditions: heads in the same class have tails spanning the same multiset of classes, and the quotient digraph is acyclic. The forward direction follows directly from a layered function; conversely, a topological ordering of the acyclic quotient digraph yields the layered labels. This reduction converts a global labelling condition into local consistency plus acyclicity, which is far easier to verify or construct.

Labelled trees as certificates. Fixing $\pi_{\mathrm{co}}(K_4^{(3)})$2 and $\pi_{\mathrm{co}}(K_4^{(3)})$3, let $\pi_{\mathrm{co}}(K_4^{(3)})$4 be the complete rooted $\pi_{\mathrm{co}}(K_4^{(3)})$5-ary tree of depth $\pi_{\mathrm{co}}(K_4^{(3)})$6, so $\pi_{\mathrm{co}}(K_4^{(3)})$7. An admissible labeling assigns labels from $\pi_{\mathrm{co}}(K_4^{(3)})$8 with no repetition along any root-to-vertex path; $\pi_{\mathrm{co}}(K_4^{(3)})$9 denotes the collection of such labelings. A ternary relation $\pi_{\mathrm{co}}(\mathbb{F})=1/2$0 holds when the main branches of $\pi_{\mathrm{co}}(\mathbb{F})=1/2$1 are precisely the depth-$\pi_{\mathrm{co}}(\mathbb{F})=1/2$2 restrictions of $\pi_{\mathrm{co}}(\mathbb{F})=1/2$3—that is, $\pi_{\mathrm{co}}(\mathbb{F})=1/2$4 "records the truncated information" of the others. Three elementary lemmas govern this relation: it restricts coherently to lower depths, its output never coincides with an input, and crucially, for every $\pi_{\mathrm{co}}(\mathbb{F})=1/2$5-tuple of admissible labelings there exists a $\pi_{\mathrm{co}}(\mathbb{F})=1/2$6 completing the relation, constructed by choosing a fresh root label from the unused pool of size at least $\pi_{\mathrm{co}}(\mathbb{F})=1/2$7.

From colorings to the host graph. Define $\pi_{\mathrm{co}}(\mathbb{F})=1/2$8 as the set of $\pi_{\mathrm{co}}(\mathbb{F})=1/2$9-multisets of elements of $\pi_{\mathrm{co}}(F_{3,2})=1/3$0 in which some member records the truncated information of the other $\pi_{\mathrm{co}}(F_{3,2})=1/3$1. A merging argument (Lemma 4) shows that any $\pi_{\mathrm{co}}(F_{3,2})=1/3$2-graph on at most $\pi_{\mathrm{co}}(F_{3,2})=1/3$3 vertices admitting a $\pi_{\mathrm{co}}(F_{3,2})=1/3$4-coloring—a vertex map into $\pi_{\mathrm{co}}(F_{3,2})=1/3$5 under which every edge's color multiset lies in $\pi_{\mathrm{co}}(F_{3,2})=1/3$6—is layered. The merging procedure iteratively identifies parts whose tree-labels agree below some depth, maintaining the invariant while strictly decreasing the part count; acyclicity of the final quotient digraph follows because a directed cycle would force a single label to appear at both the root and a deeper vertex of one admissible labeling, contradicting path-admissibility.

Now suppose $\pi_{\mathrm{co}}(F_{3,2})=1/3$7 is non-layered on $\pi_{\mathrm{co}}(F_{3,2})=1/3$8 vertices; then $\pi_{\mathrm{co}}(F_{3,2})=1/3$9 admits no $\pi_{\mathrm{co}}(F)=0$0-coloring. Construct $\pi_{\mathrm{co}}(F)=0$1 on vertex classes indexed by $\pi_{\mathrm{co}}(F)=0$2, placing an edge on any $\pi_{\mathrm{co}}(F)=0$3-tuple whose class-label multiset belongs to $\pi_{\mathrm{co}}(F)=0$4. By the completion property above, every $\pi_{\mathrm{co}}(F)=0$5-set $\pi_{\mathrm{co}}(F)=0$6 with class labels $\pi_{\mathrm{co}}(F)=0$7 has a common neighbor class $\pi_{\mathrm{co}}(F)=0$8 disjoint from $\pi_{\mathrm{co}}(F)=0$9, giving minimum codegree at least $F$0. Any copy of $F$1 in $F$2 would induce a $F$3-coloring, a contradiction. Since $F$4, the stated bound follows.

Remarks on the layered-function axioms

A short final section addresses a technical point inherited from [DLLWY]: their definition of layered 3-graphs included a third axiom (A3), requiring that two edges sharing $F$5 labels have identical label multisets. Proposition 5 shows this axiom is redundant for general $F$6: any minimal layered function automatically satisfies (A3), via a relabelling argument that merges the largest label of a violating edge pair into the smaller maximum, strictly reducing the number of distinct labels. Hence the cleaner two-axiom definition used throughout the paper is equivalent to the original.

Limitations and open questions

The paper's contribution is a qualitative dichotomy rather than a quantitative threshold. The bound $F$7 is obtained by the crude estimate $F$8 (all labelings, not merely admissible ones), and no attempt is made to bound $F$9 tightly or to determine the true order of $\pi(F)$0 for non-layered $\pi(F)$1. Two natural questions remain open. First, what is the correct growth rate of $\pi(F)$2 over non-layered $\pi(F)$3-graphs on $\pi(F)$4 vertices—is the super-exponential decay an artifact of the method? Second, for layered $\pi(F)$5-graphs with $\pi(F)$6, the paper gives no characterization of when $\pi(F)$7; the equivalence with vanishing uniform density is established only for $\pi(F)$8 via [DLLWY], and extending it to higher uniformities would require new arguments. Additionally, the characterization of zero $\pi(F)$9-degree densities for intermediate $\ell$0 ($\ell$1) beyond layeredness is not addressed.

Conclusion

This paper establishes that non-layeredness forces positive codegree Turán density, with an explicit albeit tiny lower bound, thereby proving that vanishing $\ell$2-degree Turán density implies layeredness for all uniformities and all degree parameters. For 3-graphs this completes the program initiated by Ding, Lamaison, Liu, Wang, and Yang, yielding a full structural characterization of vanishing codegree Turán density: layeredness together with vanishing uniform Turán density, and layeredness alone in the linear case. The proof technique—translating layeredness into quotient-digraph conditions certified by labelled trees, then building a high-codegree host graph excluding all small non-layered subgraphs—may be of independent interest for other forbidden-substructure problems under minimum-degree constraints.