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Vertex-colored Tur\'{a}n theorems with applications in extremal hypergraph problems

T0 review · 0 major / 1 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read The balanced bipartite 3-uniform hypergraph uniquely maximizes the ℓ₂-norm among all K₅³-free hypergraphs on n vertices for large n.

desk verdict They settle the unique extremality conjecture for the l2-norm Turán problem on K5^3 and the exact clique count in K5^3-free 3-graphs using new vertex-colored graph theorems. read the letter →

arxiv 2606.02210 v1 pith:AHTRGY3M submitted 2026-06-01 math.CO

Wanfang Chen , Jinghua Deng , Jianfeng Hou , Xizhi Liu , Yixiao Zhang This is my paper · ORCID
classification math.CO MSC 05C3505C65
keywords extremalhypergraphtheoryTuránproblemsstabilitymethodvertex-coloredgraphsK5^3ℓ2-normunique
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper confirms that the balanced complete bipartite construction is the unique extremal example for the ℓ₂-norm version of the Turán problem for the 3-uniform clique K₅³. It proves this by establishing several new results on vertex-colored graphs that forbid balanced cliques, such as bounds on edges and the ℓ₂-norm, and a sharp theorem on crossing triangles in two-colored balanced K₄-free graphs. These tools, combined with a local modification procedure in the stability method, also allow the authors to find the exact maximum number of cliques in any K₅³-free 3-uniform hypergraph on sufficiently large n vertices. A sympathetic reader would care because this resolves two open conjectures in extremal hypergraph theory.

What carries the argument

Turán-type theorems for vertex-colored graphs forbidding balanced cliques (edge bound, ℓ₂-norm bound, and sharp crossing-triangle theorem in the two-colored balanced K₄-free case), together with a local modification procedure within the stability method that reduces the problems to showing the objective function increases near the bipartite construction.

What would settle it

Constructing a K₅³-free 3-uniform hypergraph on a large number of vertices with a strictly larger ℓ₂-norm than the balanced bipartite one, or with more cliques than predicted.

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Extended reading notes

Core claim

Balogh, Clemen, and Lidický conjectured that the balanced bipartite construction is uniquely extremal for the ℓ₂-norm Turán problem for K₅³ for all sufficiently large n; we confirm this conjecture. We also determine exactly the maximum number of cliques in an n-vertex K₅³-free 3-uniform hypergraph for all sufficiently large n.

Load-bearing premise

The relevant objective function increases under suitable local changes near the bipartite construction.

Editorial extensions

If this is right

  • The balanced bipartite construction is the unique maximizer of the ℓ₂-norm for K₅³-free 3-graphs when n is large.
  • The maximum number of cliques in such hypergraphs is achieved by the balanced bipartite construction.
  • The vertex-colored results provide new tools for other colored extremal problems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The colored Turán theorems might be useful for studying other forbidden subhypergraphs beyond K₅³.
  • Similar stability approaches could resolve exact versions of Turán problems for larger cliques or different norms.
  • Testing the local modification on small n could reveal the range where the uniqueness holds.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 1 minor

Summary. The paper proves several vertex-colored Turán theorems (edge bound, ℓ₂-norm bound, and a sharp crossing-triangle theorem for two-colored balanced K₄-free graphs) and applies them, together with the stability method and a local modification procedure, to confirm that the balanced complete bipartite 3-uniform hypergraph is the unique extremal construction for the ℓ₂-norm Turán problem on K₅³ for all sufficiently large n. It also determines the exact maximum number of K₄³'s in an n-vertex K₅³-free 3-uniform hypergraph for large n, verifying the corresponding case of the Frankl–Gryaznov–Talebanfard conjecture.

Significance. The results resolve two conjectures in extremal hypergraph theory by converting the exact problems into verifiable statements about an objective function increasing under local modifications near the balanced bipartite construction. The vertex-colored theorems appear to be of independent interest and are proved in full; the reduction via stability plus local changes supplies the exact (not merely asymptotic) statements.

minor comments (1)
  1. The abstract and introduction would benefit from a brief explicit statement of the precise objective function whose monotonicity is verified by the local modification argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The derivation relies on newly proved vertex-colored Turán theorems (edge bound, ℓ₂-norm bound, crossing-triangle theorem) as independent ingredients, combined with a standard stability-method reduction to local modifications near the bipartite construction. This reduction is not self-definitional or fitted-input; it converts the exact problem into verifiable local inequalities that the paper establishes directly. No load-bearing self-citations, uniqueness theorems imported from the same authors, or ansatzes smuggled via prior work appear. The central claim therefore rests on self-contained new results rather than reducing to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Abstract-only review limits visibility into parameters and assumptions; no free parameters or invented entities are mentioned.

assumptions (1)
  • domain assumption Turán-type theorems for vertex-colored graphs forbidding balanced cliques
    Cited as main ingredients for the hypergraph applications.

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Cite this review

Pith. "Pith review of Vertex-colored Tur\'{a}n theorems with applications in extremal hypergraph problems." pith.science (2026). https://pith.science/paper/AHTRGY3M

@misc{pith2026260602210,
  author       = {Pith},
  title        = {Pith review of: Vertex-colored Tur\'an theorems with applications in extremal hypergraph problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHTRGY3M}},
  note         = {Machine review of arXiv:2606.02210}
}
abstract

Balogh, Clemen, and Lidick\'{y} proved that the $\ell_{2}$-norm Tur\'{a}n problem for $K_{5}^{3}$ is asymptotically solved by the balanced bipartite construction, and they further conjectured that this construction is uniquely extremal for all sufficiently large $n$. We confirm this conjecture. We also determine exactly the maximum number of cliques in an $n$-vertex $K_{5}^{3}$-free $3$-uniform hypergraph for all sufficiently large $n$, thereby verifying the corresponding case of a conjecture of Frankl, Gryaznov, and Talebanfard. The main ingredients are Tur\'{a}n-type theorems for vertex-colored graphs forbidding balanced cliques, including an edge bound, an $\ell_{2}$-norm bound, and a sharp crossing-triangle theorem in the two-colored balanced $K_{4}$-free case. We also use a local modification procedure within the stability method. This reduces the exact hypergraph problems to proving that the relevant objective function increases under suitable local changes near the bipartite construction.

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Reference graph

Works this paper leans on

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