An Upper Bound on the Linear Turán Number of $k$-Crowns

Abstract: A hypergraph $H$ is said to be \emph{linear} if every pair of vertices lies in at most one hyperedge. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs (also called $r$-graphs), an $r$-graph $H$ is said to be \emph{$\mathcal{F}$-free} if it contains no member of $\mathcal{F}$ as a subhypergraph. The \emph{linear Turán number} $ex_r<sup>{\mathrm{lin}}(n,\mathcal{F})$ denotes the maximum number of edges in an $\mathcal{F}$-free linear $r$-graph on $n$ vertices. The crown is a linear $3$-graph obtained from three pairwise disjoint edges by adding an edge that intersects each of them in a distinct vertex. Recently, Gyárfás, Ruszinkó, and Sárközy~[\emph{Linear Turán numbers of acyclic triple systems}, European J.\ Combin.\ (2022)] initiated the study of bounds on the linear Turán number for acyclic $3$-uniform linear hypergraphs, including that of the crown. We extend the notion of a crown by defining a $k$-crown, denoted by $C_{1,k}<sup>r$, to be a linear $r$-graph consisting of one base edge together with $k$ pairwise disjoint edges, each intersecting the base in a distinct vertex. In this paper, we establish an upper bound on $ex_r<sup>{\mathrm{lin}}(n,C_{1,k}<sup>r)$, which in particular improves the recent bound of Zhang, Broersma, and Wang~[\emph{Generalized Crowns in Linear $r$-Graphs}, Electron.\ J.\ Combin.\ (2025)] for all $r \geq 4$, without forbidding any auxiliary configuration. We also note that the cases $k\in{1,2}$ correspond to the short linear paths $P_2<sup>r$ and $P_3<sup>r$, and can be treated separately.

• The paper introduces a refined upper bound on the linear Turán number for k-crowns, generalizing previous results in 3-uniform hypergraphs.
• It employs innovative degree sequence analysis and greedy induction to derive tighter constraints without needing auxiliary forbidden structures.
• Improved bounds for the case k = r demonstrate the method’s strength, opening avenues for research into asymptotic tightness and lower bound constructions.

Upper Bounds on the Linear Turán Number for $k$-Crowns in $r$-Uniform Linear Hypergraphs

Introduction

The paper "An Upper Bound on the Linear Turán Number of $k$-Crowns" (2604.10467) is situated within the field of extremal combinatorics, specifically addressing Turán-type problems in the context of $r$-uniform linear hypergraphs. The focus is on the extremal function $ex_r^{\mathrm{lin}}(n,\mathcal{F})$, the maximal allowable edge count in a linear $r$-graph (i.e., an $r$-uniform hypergraph where every pair of vertices is contained in at most one edge) that avoids containing any member of a forbidden hypergraph family $\mathcal{F}$ as a subgraph.

This work sharpens upper bounds for configurations known as $k$-crowns, denoted $C_{1,k}^r$. A $r$0-crown is constructed as one base edge plus $r$1 pairwise disjoint edges, each intersecting the base in a unique vertex. This model generalizes previously studied instances, notably the so-called "crown" in 3-uniform hypergraphs, and encompasses the short linear paths $r$2 and $r$3 as degenerate cases for $r$4.

Main Contributions

Definition and Generalization

The paper formalizes the $r$5-crown $r$6 for $r$7, extending earlier work for $r$8 to arbitrary $r$9, and provides a unified treatment for these configurations within linear $k$0-graphs. The main problem is to determine, or tightly bound, $k$1: the maximum number of edges in a linear $k$2-graph on $k$3 vertices that contains no $k$4 as a subgraph.

Improved Upper Bounds

The central theorem of the paper states: $k$5 where $k$6 is the number of vertices of degree at least $k$7. This bound is obtained under the sole assumption of $k$8-freeness, in contrast to previous results that required additional forbidden auxiliary configurations.

In the important case $k$9 (the "full crown"), this result is shown to strictly improve the best known upper bound established by Zhang, Broersma, and Wang ("Generalized Crowns in Linear $r$0-Graphs", Electron. J. Combin., 2025), which was: $r$1 for $r$2. The new bound retains the same degree threshold for the removal of vertices but achieves a uniformly smaller leading coefficient, guaranteeing a stronger result for all sufficiently large $r$3 without supplementary forbidden structures.

Additionally, the cases $r$4 (corresponding to linear paths $r$5) are also subsumed, albeit as degenerate crowns, confirming that the main argument holds formally in these regimes.

Structural and Methodological Refinements

The proof technique advances the state of the art through the following methodological innovations:

• The argument exploits degree sequences and carefully constructed induction/greedy procedures, applying a double counting method to leverage the distribution of low- and high-degree vertices.
• The result yields a "localized" refinement, connecting to recent research on weighted extremal bounds in hypergraphs (see, e.g., [adak2025vertex], [malec2023localized]).

A key technical lemma shows that if a base edge contains $r$6 vertices of sufficiently high degree, a $r$7-crown must appear. The absence of a crown therefore implies that every edge contains a vertex with degree below a certain threshold, tightly controlling the aggregate degree and, consequently, the number of edges.

Comparison with Existing Results

The result sharpens and generalizes the following prior bounds:

• For the case $r$8, $r$9 arbitrary, the upper bound matches $ex_r^{\mathrm{lin}}(n,\mathcal{F})$0, as previously established for trees with four edges in [adak2026bounds].
• For $ex_r^{\mathrm{lin}}(n,\mathcal{F})$1, the coefficient in the new upper bound is strictly less than that of the earlier best-known bound, even though the earlier proof required forbidding additional structures ($ex_r^{\mathrm{lin}}(n,\mathcal{F})$2).

Notably, the proof dispenses with complicated structural arguments concerning auxiliary configurations, relying strictly on the $ex_r^{\mathrm{lin}}(n,\mathcal{F})$3-free condition and degree considerations.

Implications and Directions for Future Research

The theoretical implications of this work are significant regarding the extremal theory of linear hypergraphs. By delivering a tighter bound using simpler hypotheses, the work quantifies the true extremal limitations imposed by crown-type configurations and opens several questions:

• For fixed $ex_r^{\mathrm{lin}}(n,\mathcal{F})$4 and $ex_r^{\mathrm{lin}}(n,\mathcal{F})$5, is the upper bound asymptotically tight?
• Can sharper or matching lower bounds be constructed for $ex_r^{\mathrm{lin}}(n,\mathcal{F})$6-free linear $ex_r^{\mathrm{lin}}(n,\mathcal{F})$7-graphs, answering whether the degree-based analysis is optimal?
• Is it possible to adapt the degree-sum and localization framework to other families of forbidden acyclic substructures, or to even more general extremal problems in sparse hypergraphs?

From a practical standpoint, these bounds inform limits on the structure and density of code complexes, information networks, and related discrete structures under specified local intersection constraints.

Conclusion

This paper establishes new, strictly improved upper bounds on the linear Turán number for $ex_r^{\mathrm{lin}}(n,\mathcal{F})$8-crowns in $ex_r^{\mathrm{lin}}(n,\mathcal{F})$9-uniform linear hypergraphs, employing refined degree-sequence and localization methodologies. By providing a bound that avoids additional forbidden configurations and yields better coefficients for $r$0, the result advances the extremal theory of sparse hypergraphs. The methods and insights presented invite further investigation into tightness and generalization of degree-localized extremal results in hypergraph theory.