Codegree Turán Density

• Codegree Turán density is a local extremal parameter in k-uniform hypergraphs that measures the asymptotic minimum codegree required to force a forbidden subgraph.
• It refines classical Turán density by integrating local codegree conditions, with sharp examples demonstrated in 3-uniform cases such as K4^(3) and specific tight cycles.
• The study employs algebraic constructions, supersaturation arguments, and link-graph techniques to tackle longstanding combinatorial challenges in hypergraph theory.

A codegree Turán density is a local extremal parameter for uniform hypergraphs, describing the asymptotic minimum codegree threshold that forces the appearance of a fixed forbidden hypergraph. For a fixed $k$-uniform hypergraph %%%%1%%%%, its codegree Turán density $\gamma(F)$ is the infimum $\gamma\in[0,1]$ such that every sufficiently large $n$-vertex $k$-graph $H$ with minimum codegree at least $(\gamma+o(1))n$ must contain $F$ as a (possibly homomorphic) subgraph. The limiting density exists and refines ordinary Turán density, offering a local–to–global perspective that is central to modern extremal hypergraph theory (Piga et al., 2024, Piga et al., 2023, Lo et al., 2012).

1. Formal Definition and Basic Properties

Let $H$ be a $k$-uniform hypergraph on $n$ vertices. For each $(k-1)$-set $S\subset V(H)$, the codegree $d_H(S)$ is the number of edges containing $S$. Define the minimum codegree: $\delta_{k-1}(H) = \min_{S\in V(H)^{(k-1)}} d_H(S).$ The codegree Turán number for $F$ is: $$\mathrm{ex}_{k-1}(n,F) = \max\{\,\delta_{k-1}(H)\ :\ H\text{ is }F\text{-free},\ |V(H)| = n\,\}.$$ The (asymptotic) codegree Turán density is then: $$\gamma(F) = \lim_{n\to\infty} \frac{\mathrm{ex}_{k-1}(n,F)}{n}.$$ This limit always exists (Piga et al., 2023, Lo et al., 2012). It satisfies $0 \leq \gamma(F) \leq \pi(F)$ where $\pi(F)$ is the classical Turán density. For $k=2$ (graphs), one always has $\gamma(F)=\pi(F)$, but for $k\geq3$ and nontrivial $F$, $\gamma(F)$ can differ sharply from $\pi(F)$ (Lo et al., 2012).

2. Paradigmatic Examples and Explicit Values

While general determination is notoriously difficult, several explicit cases have been resolved for $k=3$:

• For the complete 3-graph $K_4^{(3)}$, Czygrinow–Nagle conjecture $\gamma(K_4^{(3)})=1/2$ (Piga et al., 2024, Lo et al., 2012).
• For $K_4^{(3)-}$ (the unique 3-edge subgraph of $K_4$), $\gamma(K_4^{(3)-})=1/4$ (Lo et al., 2012).
• For the tight cycle $C_\ell$ of length $\ell$ in the 3-uniform case:
  • If $3$ divides $\ell$, then $C_\ell$ is tripartite and $\gamma(C_\ell)=0$.
  • For $\ell\not\equiv0\pmod3$ and $\ell\in\{10,13,16\}$ or $\ell \geq 19$, $\gamma(C_\ell)=1/3$ (Piga et al., 2024, Ma, 2024).
  • For $C_{11}$, Ma confirmed $\gamma(C_{11})=1/3$ (Ma, 2024).
• For tight cycles minus one edge, $C_\ell^{-}$, for all $\ell\geq5$, $\gamma(C_\ell^{-})=0$ (Piga et al., 2022).
• The Fano plane (unique 7-vertex 3-graph with every pair in an edge): $\gamma=\frac{1}{2}$ (Lo et al., 2012).

3. Structural and Combinatorial Frameworks

Proof techniques for codegree Turán densities apply a blend of local expansion, link graphs, and the auxiliary digraph method, especially for tight cycles:

• For 3-uniform tight cycles with $\ell\not\equiv0\pmod3$, the extremal construction is a balanced 3-partite hypergraph with edges corresponding to the sum of indices (mod 3). This achieves $\delta_2(H)\geq n/3-1$, yet excludes any non-tripartite tight cycle (Piga et al., 2024).
• Embedding arguments typically build around $\mathsf{K}_4^{-}$ extensions: every edge is forced inside a small structure unless a forbidden cycle appears, leading to the emergence of homomorphic images of constrained $C_\ell$ for sufficiently large minimum codegree (Piga et al., 2024, Ma, 2024).

For tight cycles minus one edge, the codegree threshold drops to zero; thus, for any positive $\alpha$, every large enough 3-graph with codegree exceeding $\alpha n$ must contain $C_\ell^{-}$, indicating these are inclusion-minimal for positive codegree density (Piga et al., 2022).

For complete $r$-graphs $K_t^r$ ($r\geq3$), the density satisfies: $$1 - c_2 \frac{\ln t}{t^{r-1}} \leq \gamma(K_t^r) \leq 1 - c_1 \frac{\ln t}{t^{r-1}},$$ with explicit $c_1,c_2=r$-dependent constants (Lo et al., 2018), establishing sharp asymptotics as $t\to\infty$.

4. Accumulation Points and Density of Codegree Values

Piga–Schülke (Piga et al., 2023), Li–Liu–Schülke–Sun (Li et al., 19 Feb 2025), and Lo–Markström (Lo et al., 2012) collectively established that the set of all codegree Turán densities,

$$\Gamma^{(k)} := \{\,\gamma(F): F~\text{is a}~k\text{-graph}\,\} \subset [0,1),$$

is extremely rich:

• For all $k\geq3$ and integers $r\geq1$, $(r-1)/r$ is an accumulation point of $\Gamma^{(k)}$ (Li et al., 19 Feb 2025).
• In particular, there is no "jump" at $0$: for every $\varepsilon>0$, there exists $F$ with $0<\gamma(F)<\varepsilon$ (Piga et al., 2023).
• In contrast, for classical Turán density, Erdős proved there is a gap $(0, k!/k^k)$ containing no density values.

Table: Comparative thresholds

Invariant	Smallest nonzero possible value	Known distribution
Turán density $\pi(F)$	$k!/k^k$ (Erdős, 1964)	Not dense near 0
Codegree $\gamma(F)$	$0$ (Piga–Schülke, 2023)	Dense in $[0,1)$ for $k>2$

5. Connections to Other Notions

There are profound links between codegree Turán density and other extremal parameters:

• Uniform Turán density: For 3-graphs, vanishing codegree density implies vanishing uniform density; the converse holds for "layered" graphs but fails in general (Ding et al., 2024, Ding et al., 2023).
• $\ell$-degree Turán density: The codegree case is $\ell=k-1$, but nontrivial results hold for all $1<\ell<k$ (Lo et al., 2012).
• Codegree squared extremal function: The maximum sum of squared codegrees (the $\ell_2$ norm of the codegree vector) displays different thresholds, with $\pi_2(H)=0$ even for matchings, stars, and cycles (Balogh et al., 2021).

6. Critical Proof Techniques and Constructions

• Algebraic constructions: Balanced multipartite hypergraphs partitioned by residue classes or group labels underlie extremal examples (Piga et al., 2024, Li et al., 19 Feb 2025).
• Supersaturation and blow-up: Once a codegree threshold is exceeded, not only single, but linearly many copies of $F$ must appear. Blow-up lemmas ensure that codegree density is invariant under graph blow-ups (Piga et al., 2023, Piga et al., 2022).
• Link-graph and Ramsey-theoretic analysis: Extension from local codegree constraints to global structure hinges on embedding via dense links and intersection patterns (Ding et al., 2024).

7. Open Problems and Recent Advances

• Determining $\gamma(F)$ for $K_4^{(3)}$ remains open; conjectured value is $1/2$ (Lo et al., 2012).
• For 3-uniform tight cycles, only $\ell=7$ remains unresolved for the nonzero, nontripartite case (Ma, 2024).
• The density and possible irrational accumulation points for $\Gamma^{(k)}$ are conjectured, but classification is incomplete (Li et al., 19 Feb 2025).
• Classifying $k$-graphs $F$ with $\gamma(F)=0$ is both structurally and combinatorially open: ordered link and bipartition frameworks provide significant progress for tight cycles and zycles, but a unifying theory remains elusive (Sarkies, 30 Mar 2025).

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In sum, codegree Turán density is a deep, robust local extremal invariant for uniform hypergraphs, sharply distinguishing itself from global (edge) density: it admits dense value sets, minimal positive cases, and features intricate combinatorial behavior. The study of codegree thresholds not only solves longstanding conjectures for tight cycles and their relatives, but also enriches the theoretical toolkit for extremal combinatorics, blending classical algebraic, probabilistic, and structural techniques (Piga et al., 2024, Piga et al., 2023, Ma et al., 28 Dec 2025, Li et al., 19 Feb 2025, Piga et al., 2022, Lo et al., 2012).