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arxiv: 2606.22828 · v1 · pith:NZNZYOFAnew · submitted 2026-06-22 · 🧮 math.CO

Tur\'an numbers of 4-uniform tight even cycles minus one edge

Wanfang Chen , Jianfeng Hou , Xizhi Liu , Yixiao Zhang , Hongbin Zhao This is my paper

Pith reviewed 2026-06-26 08:10 UTC · model grok-4.3

classification 🧮 math.CO
keywords Turán numbers4-uniform hypergraphstight cyclesodd-bipartite hypergraphsextremal graph theorystabilityhypergraph Turán problem
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The pith

For large n the extremal 4-uniform hypergraphs avoiding a tight even cycle of length 4k+2 minus one edge are the complete odd-bipartite 4-graphs.

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

The paper proves that for every fixed k at least 1 and all sufficiently large n, the largest number of edges in a 4-uniform hypergraph without a copy of the tight cycle C_{4k+2}^4 minus one edge is attained precisely by the complete odd-bipartite 4-graphs. This construction is shown to be the unique extremal example and yields the Turán density 1/2 for the full cycle C_{4k+2}^4 when k is at least 2, together with a stability statement that any hypergraph with nearly that many edges must be close in structure to an odd-bipartite example. The result recovers the known density for the 4-uniform expanded triangle as the k=1 case and improves an earlier density theorem that held only for large k. A reader cares because the work fixes the asymptotic maximum edge density for an infinite family of forbidden subhypergraphs and supplies the structural information needed to understand near-extremal examples.

Core claim

For every integer k ≥ 1 and sufficiently large n, the extremal construction for the Turán number of the 4-uniform tight cycle of length 4k+2 minus one edge is a complete odd-bipartite 4-graph. In particular the Turán density of C_{4k+2}^4 is 1/2 for all k ≥ 2 together with the corresponding stability result.

What carries the argument

The complete odd-bipartite 4-graph, the unique maximum-edge construction shown to avoid every copy of C_{4k+2}^{4-} while attaining the stated density.

If this is right

  • The Turán density of every C_{4k+2}^4 equals 1/2 for k ≥ 2.
  • Any 4-uniform hypergraph with edge count within o(n^4) of the extremal number must be edit-close to a complete odd-bipartite 4-graph.
  • The same extremal construction works for the expanded-triangle case, recovering and extending the Frankl–Keevash–Sudakov theorem.
  • Stability transfers to the full cycle problem, so near-maximal hypergraphs without C_{4k+2}^4 are also close to odd-bipartite examples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The stability result may be strong enough to determine the exact Turán number for every n once the threshold is made explicit.
  • The same odd-bipartite construction could serve as the extremal example for other even-length tight cycles in higher uniformities.
  • Removing one edge from the cycle appears to be the minimal change that makes the density drop from its conjectured value to exactly 1/2.

Load-bearing premise

That n is large enough for stability and supersaturation arguments to rule out all other constructions.

What would settle it

An explicit 4-uniform hypergraph on n vertices (n larger than the paper's implicit threshold for a given k) that contains more edges than any complete odd-bipartite 4-graph yet still contains no copy of C_{4k+2}^{4-}.

Figures

Figures reproduced from arXiv: 2606.22828 by Hongbin Zhao, Jianfeng Hou, Wanfang Chen, Xizhi Liu, Yixiao Zhang.

Figure 1
Figure 1. Figure 1: From left to right: the 4-uniform expanded triangle C 4 3 , the 4-uniform tight cycle of length 6 minus one edge C 4− 6 , and the complete odd-bipartite 4-graph B odd 4 (n). Our first main theorem gives the density and stability statement for the tight cycles in this congruence class. The extremal model is again the complete odd-bipartite construction. Theorem 1.1. Let k ≥ 2 be an integer. Then π(C 4 4k+2)… view at source ↗
read the original abstract

For every integer $k \ge 1$ and sufficiently large $n$, we show that the extremal construction for the Tur\'{a}n number of the $4$-uniform tight cycle of length $4k+2$ minus one edge is a complete odd-bipartite $4$-graph. In particular, since $C_{6}^{4-}$ contains the $4$-uniform expanded triangle as a subgraph, our result extends that of Frankl and Keevash--Sudakov on the Tur\'an density and the Tur\'{a}n number of the $4$-uniform expanded triangle. We also show that the Tur\'{a}n density of $C_{4k+2}^{4}$ is $1/2$ for all integers $k \ge 2$, and establish the corresponding stability result. This strengthens the result of Sankar on the Tur\'{a}n density of $C_{4k+2}^{4}$ which holds only for all sufficiently large $k$.

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

2 major / 1 minor

Summary. The paper proves that for every integer k ≥ 1 and all sufficiently large n, the unique extremal 4-uniform hypergraph on n vertices avoiding a tight 4-cycle of length 4k+2 minus one edge is the complete odd-bipartite 4-graph. As a consequence it obtains the Turán density π(C_{4k+2}^4) = 1/2 together with stability for every k ≥ 2, extending the Frankl–Keevash–Sudakov result on the expanded triangle and improving Sankar’s theorem (which required k sufficiently large).

Significance. If correct, the result supplies an exact extremal construction and density for an infinite family of 4-uniform even cycles (minus an edge) that was previously known only asymptotically in k. The stability statement is a standard strengthening in the area and the reduction to the odd-bipartite construction is the central new contribution.

major comments (2)
  1. [Introduction] Introduction, paragraph following the statement of the main theorem: the claim that the result strengthens Sankar by removing the “sufficiently large k” hypothesis rests on the existence of N(k) for every fixed k ≥ 2. The stability and supersaturation arguments invoked to obtain this N(k) necessarily pass through the hypergraph removal lemma (or an iterative supersaturation step) whose quantitative bounds depend on the length 4k+2; no explicit bound or growth rate for N(k) is supplied, leaving open whether the improvement over Sankar is uniform in the stated sense.
  2. [Main theorem statement] The proof that the complete odd-bipartite 4-graph is extremal for C_{4k+2}^{4-} (the minus-one-edge version) is used to deduce the density result for the full cycle C_{4k+2}^4. If the stability argument for the minus-one-edge problem contains a k-dependent error term that is not controlled uniformly, the deduction that π(C_{4k+2}^4) = 1/2 for every k ≥ 2 would require an additional limiting argument that is not indicated in the abstract.
minor comments (1)
  1. The abstract cites the Frankl–Keevash–Sudakov and Sankar results but does not give their precise statements or reference numbers; adding these in the introduction would clarify the exact strengthening.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Introduction] Introduction, paragraph following the statement of the main theorem: the claim that the result strengthens Sankar by removing the “sufficiently large k” hypothesis rests on the existence of N(k) for every fixed k ≥ 2. The stability and supersaturation arguments invoked to obtain this N(k) necessarily pass through the hypergraph removal lemma (or an iterative supersaturation step) whose quantitative bounds depend on the length 4k+2; no explicit bound or growth rate for N(k) is supplied, leaving open whether the improvement over Sankar is uniform in the stated sense.

    Authors: The main result is stated existentially: for every fixed integer k ≥ 1 there exists N = N(k) such that the unique extremal hypergraph on n > N vertices is the complete odd-bipartite 4-graph. This removes Sankar’s hypothesis that k itself must exceed some absolute constant, which is the strengthening claimed in the manuscript. The dependence of the quantitative bounds (via the removal lemma) on the cycle length 4k+2 is standard and does not invalidate the per-k existential statement. The manuscript makes no claim of a uniform N independent of k. revision: no

  2. Referee: [Main theorem statement] The proof that the complete odd-bipartite 4-graph is extremal for C_{4k+2}^{4-} (the minus-one-edge version) is used to deduce the density result for the full cycle C_{4k+2}^4. If the stability argument for the minus-one-edge problem contains a k-dependent error term that is not controlled uniformly, the deduction that π(C_{4k+2}^4) = 1/2 for every k ≥ 2 would require an additional limiting argument that is not indicated in the abstract.

    Authors: For each fixed k ≥ 2 the stability theorem for the minus-one-edge problem supplies an error term that may depend on k. The Turán density π(C_{4k+2}^4) is the limit of ex(n, C_{4k+2}^4)/binom(n,4) as n → ∞ with k held fixed; any k-dependent error therefore disappears in this limit. The deduction therefore requires no additional limiting process in which k tends to infinity, and the abstract statement for all k ≥ 2 is justified by fixing k first. revision: no

Circularity Check

0 steps flagged

No circularity; extends independent external theorems via standard stability

full rationale

The derivation invokes stability and supersaturation methods applied to the new forbidden subgraph C_{4k+2}^{4-}, extending results of Frankl-Keevash-Sudakov (on the expanded triangle) and Sankar (on large cycles) without any reduction of the central claim to a self-defined quantity, fitted parameter, or self-citation chain. No equations or steps in the provided abstract or description equate the extremal construction or density 1/2 to inputs by construction; the 'sufficiently large n' threshold is a standard non-quantified asymptotic hypothesis common to the field and does not create circularity. The result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a proof in extremal combinatorics and introduces no new free parameters, invented entities, or non-standard axioms beyond the usual definitions of uniform hypergraphs, tight cycles, and Turán numbers.

axioms (1)
  • standard math Standard definitions and basic properties of 4-uniform hypergraphs, tight cycles, and the Turán number ex(n, F).
    Invoked throughout the abstract when stating the extremal construction and density.

pith-pipeline@v0.9.1-grok · 5718 in / 1334 out tokens · 23690 ms · 2026-06-26T08:10:49.820399+00:00 · methodology

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Reference graph

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