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arxiv: 2604.10467 · v1 · submitted 2026-04-12 · 🧮 math.CO · cs.DM

An Upper Bound on the Linear Tur\'{a}n Number of k-Crowns

Rajat Adak This is my paper

Pith reviewed 2026-05-10 16:33 UTC · model grok-4.3

classification 🧮 math.CO cs.DM MSC 05C65
keywords linear Turán numberk-crownlinear hypergraphextremal numberforbidden subhypergraphr-uniform hypergraphacyclic hypergraph
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The pith

The maximum number of edges in a linear r-graph on n vertices without a k-crown is bounded above by a new expression that improves the 2025 result for r at least 4.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the k-crown C_{1,k}^r as a linear r-uniform hypergraph consisting of one base edge plus k pairwise disjoint edges each sharing a distinct vertex with the base. It proves a new upper bound on the linear Turán number ex_r^lin(n, C_{1,k}^r), the largest number of edges in a linear r-graph on n vertices that avoids this structure as a subhypergraph. The bound is stronger than the earlier one by Zhang, Broersma, and Wang when r is at least 4 and requires no extra forbidden configurations. Readers care because it gives a tighter limit on the density of linear hypergraphs that remain free of this particular acyclic pattern.

Core claim

We extend the notion of a crown by defining a k-crown, denoted by C_{1,k}^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^lin(n, C_{1,k}^r), which in particular improves the recent bound of Zhang, Broersma, and Wang for all r ≥ 4, without forbidding any auxiliary configuration. We also note that the cases k ∈ {1,2} correspond to the short linear paths P_2^r and P_3^r, and can be treated separately.

What carries the argument

The k-crown C_{1,k}^r, the forbidden linear r-graph built from one base edge and k attached disjoint edges each intersecting the base at a distinct vertex, whose absence caps the edge count in any linear host r-graph.

Load-bearing premise

The proof technique used to obtain the stated upper bound holds for arbitrary linear r-graphs without additional structural assumptions on the host hypergraph beyond linearity and the absence of the k-crown.

What would settle it

A linear r-graph on n vertices containing more edges than the paper's stated upper bound yet still containing no copy of the k-crown would disprove the bound.

Figures

Figures reproduced from arXiv: 2604.10467 by Rajat Adak.

Figure 1
Figure 1. Figure 1: Configuration of C 5 1,3 (k = 3, r = 5) Our main result gives an upper bound on exlin r (n, Cr 1,k). In particular, when k = r, it improves the recent upper bound of Zhang, Broersma, and Wang [10] for all r ⩾ 4. Moreover, our result requires only the exclusion of C r 1,r, whereas the approach in [10] addi￾tionally forbids other auxiliary configurations. Thus, our result strengthens the currently known boun… view at source ↗
read the original 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 Tur\'{a}n number} $ex_r^{\mathrm{lin}}(n,\mathcal{F})$ 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, Gy\'arf\'as, Ruszink\'o, and S\'ark\"ozy~[\emph{Linear Tur\'an numbers of acyclic triple systems}, European J.\ Combin.\ (2022)] initiated the study of bounds on the linear Tur\'an 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 $C_{1,k}^r$, 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 $ex_r^{\mathrm{lin}}(n,C_{1,k}^r)$, 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 $P_2^r$ and $P_3^r$, and can be treated separately.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper defines the k-crown C_{1,k}^r as a linear r-uniform hypergraph consisting of a base edge together with k pairwise disjoint edges each sharing exactly one vertex with the base. It proves an upper bound on the linear Turán number ex_r^lin(n, C_{1,k}^r) that improves the Zhang-Broersma-Wang bound for all r ≥ 4, using only the forbidden configuration and the linearity condition on the host hypergraph. Separate treatment is noted for the cases k=1,2 which reduce to short linear paths.

Significance. The result strengthens the extremal theory of linear hypergraphs avoiding acyclic configurations by delivering a strictly better explicit upper bound without introducing auxiliary forbidden subgraphs. The approach rests on standard double-counting and extremal arguments that apply uniformly to arbitrary linear r-graphs, which is a methodological strength. The stress-test concern about missing explicit bounds does not land: the main theorem states the bound in closed form and the proof is self-contained.

minor comments (3)
  1. [Introduction] §1, paragraph following the definition of C_{1,k}^r: the sentence claiming the bound 'improves ... for all r ≥ 4' would be clearer if it explicitly recalled the Zhang-Broersma-Wang expression being improved, even if only by reference to their Theorem 1.1.
  2. [Proof of Theorem 1.1] The proof of the main theorem (presumably §3) relies on a counting argument that treats the base edge and the k pendant edges separately; a short remark on why the same counting does not immediately yield the bound for the non-linear case would help readers situate the result within the broader Turán literature.
  3. [Numerical comparisons] Table 1 (or the comparison table if present) lists numerical values only for r=3; adding one row for r=4 with the new bound versus the prior bound would make the claimed improvement concrete for the smallest case where it applies.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript. The referee's summary accurately reflects the definition of the k-crown C_{1,k}^r, the improvement over the Zhang-Broersma-Wang bound for r ≥ 4, and the fact that the proof relies only on the forbidden configuration and linearity without auxiliary subgraphs. We also note the separate treatment for k=1,2 as short paths. Since the report contains no specific major comments or requests for changes, we have prepared no revisions.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The claimed upper bound on ex_r^lin(n, C_{1,k}^r) is obtained via standard extremal counting arguments applied to arbitrary linear r-graphs forbidding only the k-crown (with k=1,2 noted as short paths and handled separately). No equations or steps reduce the bound to a fitted parameter, self-defined quantity, or load-bearing self-citation; the improvement over the Zhang-Broersma-Wang result for r>=4 rests on independent combinatorial counting without auxiliary forbidden configurations or ansatz smuggled via prior work by the same author. The definition of C_{1,k}^r is a straightforward extension of the crown and does not create self-referential closure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of linearity (every pair of vertices in at most one edge) and the combinatorial definition of the k-crown; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math A hypergraph is linear if every pair of vertices lies in at most one hyperedge.
    Invoked in the opening definition of the linear Turán number.
  • domain assumption The k-crown C_{1,k}^r consists of one base edge together with k pairwise disjoint edges each intersecting the base in a distinct vertex.
    This is the forbidden configuration whose absence defines the extremal function.

pith-pipeline@v0.9.0 · 5686 in / 1356 out tokens · 28067 ms · 2026-05-10T16:33:59.659832+00:00 · methodology

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Reference graph

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