--- title: Codegree Turán Density url: https://www.emergentmind.com/topics/codegree-turan-density type: topic --- # Codegree Turán Density 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 $F$, 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 [2408.02588], [2307.02876], [1210.5726]. ## 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 [2307.02876], [1210.5726]. 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 $\pi(F)$ [1210.5726]. ## 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 $\gamma(K_4^{(3)})=1/2$ [2408.02588], [1210.5726]. - For $K_4^{(3)-}$ (the unique 3-edge subgraph of $K_4$), $\gamma(K_4^{(3)-})=1/4$ [1210.5726]. - 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 $\ell \geq 19$, $\gamma(C_\ell)=1/3$ [2408.02588], [2409.02765]. - For $C_{11}$, Ma confirmed $\gamma(C_{11})=1/3$ [2409.02765]. - For tight cycles minus one edge, $C_\ell^{-}$, for all $\ell\geq5$, $\gamma(C_\ell^{-})=0$ [2211.12721]. - The Fano plane (unique 7-vertex 3-graph with every pair in an edge): $\gamma=\frac{1}{2}$ [1210.5726]. ## 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 [2408.02588]. - 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 [2408.02588], [2409.02765]. 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 [2211.12721]. 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 [1801.01393], establishing sharp asymptotics as $t\to\infty$. ## 4. Accumulation Points and Density of Codegree Values Piga–Schülke [2307.02876], Li–Liu–Schülke–Sun [2502.13485], and Lo–Markström [1210.5726] 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 $\Gamma^{(k)}$ [2502.13485]. - In particular, there is no "jump" at $0$: for every $\varepsilon>0$, there exists $F$ with $0<\gamma(F)<\varepsilon$ [2307.02876]. - 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 [2407.08771], [2312.02879]. - **$\ell$-degree Turán density**: The codegree case is $\ell=k-1$, but nontrivial results hold for all $1<\ell