Strong non-principality of positive codegree Turán density

Abstract: The \emph{minimum positive codegree} $δ<sup>+_{k-1}(G)$ of a $k$-graph $G$ is the minimum, over all $(k-1)$-sets that lie in at least one edge, of the number of edges containing that set. The \emph{positive codegree Turán density} of a $k$-graph family $\mathcal{F}$ is the asymptotically maximum value of $δ<sup>+_{k-1}(G)/n$ over all $\mathcal{F}$-free $k$-graphs $G$ with $n\to\infty$ vertices. In this note, we establish a strong version of non-principality with respect to this density by proving that for every $k\ge3$ there exist two $k$-graphs $F_1$ and $F_2$ such that $$ 0&lt;γ^+(F_1, F_2) &lt; \min{γ^+(F_1), γ^+(F_2)}. $$

• The paper demonstrates that for every k ≥ 3, there exist explicit k-graphs F1 and F2 for which the joint positive codegree density is strictly lower than the densities of each forbidden subgraph.
• The paper introduces constructive recursive methods and explicit hypergraph constructions, such as Q2^k and D_{k+1}^k, to establish sharp density bounds.
• The paper’s findings refine extremal hypergraph theory by showing a non-principality gap in positive codegree Turán density and suggesting new avenues for algorithmic applications.

Strong Non-Principality of Positive Codegree Turán Density

Introduction

The paper "Strong non-principality of positive codegree Turán density" (2606.20494) addresses the extremal properties of $k$-uniform hypergraphs with respect to minimum positive codegree Turán density. The study investigates whether for each $k \ge 3$, there exist pairs of forbidden $k$-graphs such that the positive codegree Turán density of their joint exclusion is strictly less than the minimum of their individual densities—a strong non-principality. The research extends foundational results on Turán-type densities, particularly in the positive codegree context, which encompasses all $(k-1)$-sets appearing in at least one edge.

Context and Definitions

Given a $k$-uniform hypergraph $G$, the minimum positive codegree $\delta^+_{k-1}(G)$ measures, among $(k-1)$-sets in at least one edge, the smallest number of edges intersecting such a set. The positive codegree Turán density $\gamma^+(\mathcal{F})$ for a family $\mathcal{F}$ of $k \ge 3$0-graphs is defined as the asymptotic maximal normalized minimum positive codegree over all $k \ge 3$1-free $k \ge 3$2-graphs as $k \ge 3$3. This parameter, introduced to make partite constructions relevant in codegree extremal theory, generalizes classical Turán density to local edge containment, highlighting subtler forbidden subgraph effects.

Principal versus non-principal families delineate whether the density for the joint exclusion of a family is strictly less than exclusion of any single member. While such phenomena are not present for graphs ($k \ge 3$4), they occur in hypergraphs ($k \ge 3$5), as established for edge and classical codegree densities. The present paper establishes this phenomenon for positive codegree density with explicit constructions.

Main Results and Technical Claims

The central theorem asserts that for every $k \ge 3$6, there exist $k \ge 3$7-graphs $k \ge 3$8 and $k \ge 3$9 such that

$k$0

This demonstrates strong non-principality: the joint exclusion of $k$1 and $k$2 forces strictly lower positive codegree density than either forbidden alone. The proof is constructive, relying on recursive combinatorial constructions and tight density analysis.

Construction and Bounds

• Explicit Hypergraph Construction: The paper introduces $k$3 and $k$4 as candidate forbidden $k$5-graphs. $k$6 is based on a $k$7-graph $k$8, augmented with edges ensuring structural constraints. $k$9 connects a fixed $(k-1)$0-set to all pairs in an $(k-1)$1-set.
• Sharp Density Values: It is shown that $(k-1)$2 and $(k-1)$3. By examining balanced complete multipartite graphs, it is established that $(k-1)$4, while detailed structural analysis and inductive arguments relate the upper bound to $(k-1)$5 for small $(k-1)$6.
• Non-Principal Gap: The result $(k-1)$7 guarantees a gap, confirming non-principality in the positive codegree setting.

Auxiliary Results

• Recursive Codegree Construction: The paper defines $(k-1)$8 recursively, where $(k-1)$9 and $k$0 is non-$k$1-colourable, generalizing Turán extremal constructions.
• Structural Lemmas: Technical claims establish bounds on shadows, degree properties, and partition arguments that underpin the extremal densities.

Numerical Results and Explicit Claims

The bold, contradictory claim is that for two explicit graphs $k$2 and $k$3, the positive codegree Turán density for their intersection is strictly smaller than either of their individual densities, i.e.,

$k$4

This is proved via explicit construction and combinatorial analysis. The lower bound of $k$5 and the explicit upper gap, for all $k$6, are strong numerical results.

Implications and Future Directions

Practically, this result informs the limits of local codegree constraints in large hypergraph constructions, especially for algorithms reliant on extremal combinatorics. Theoretically, it strengthens the landscape of hypergraph Turán-type densities by demonstrating new classes of forbidden families exhibiting non-principality, further delineating complexity between edge, codegree, and positive codegree parameters.

Future developments could include:

• Characterization of all pairs (or larger families) exhibiting strong non-principality for positive codegree densities.
• Extension to other local density parameters, such as generalized $k$7-degree or to random hypergraph models.
• Algorithmic applications for extremal hypergraph enumeration, local property testing, or bounding thresholds in random hypergraph processes.

Conclusion

The paper rigorously establishes strong non-principality for the positive codegree Turán density in $k$8-uniform hypergraphs, providing explicit constructions and sharp density bounds. These results further establish the nuanced structure of extremal hypergraph theory and broaden the understanding of local density phenomena in combinatorics. The implications suggest further combinatorial and algorithmic exploration in Turán-type extremal problems for complex hypergraph families.