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arxiv: 2605.15105 · v1 · submitted 2026-05-14 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

Uniform Tur\'an densities of k-uniform hypergraphs

Hao Lin , Guowei Sun , Guanghui Wang , Wenling Zhou
Authors on Pith no claims yet

Pith reviewed 2026-05-15 03:15 UTC · model grok-4.3

classification 🧮 math.CO
keywords k-uniform hypergraphsuniform Turán densitypalette frameworkextremal hypergraph theoryTurán densitynon-principal familieshypergraph colorability
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The pith

For any family of k-graphs, the (k-2)-uniform Turán density equals the palette Turán density.

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

The paper develops a palette framework for the (k-2)-uniform Turán density of families of k-uniform hypergraphs. It proves that this density is always equal to the corresponding palette Turán density. The equivalence reduces exact computations to a palette-homomorphism problem and removes the need for the hypergraph regularity method. Using the framework, the authors determine six explicit density values that hold for single k-graphs across all k >= 3 and r >= 2, and they exhibit the first pairs of k-graphs whose joint density is strictly smaller than the minimum of the separate densities.

Core claim

For every family F of k-graphs, π_{k-2}(F) equals the palette Turán density. The palette classification tools reduce the problem to determining the maximum density of a k-graph that satisfies prescribed colorability constraints on its (k-2)-subsets while avoiding the members of F. This yields the exact values (r-1)/r, (r-1)^2/r^2, (r-1)/(2r), (k-1)^k/k^k, 4(k-2)^{k-2}/k^k, and 4(k-2)^{k-2}/(3k^k) as (k-2)-uniform Turán densities of individual k-graphs, together with the existence of k-graphs F1 and F2 such that π_{k-2}({F1,F2}) < min{π_{k-2}(F1), π_{k-2}(F2)}.

What carries the argument

The palette framework, which encodes colorability constraints on the (k-2)-subsets of a k-graph and reduces density questions to the maximum size of large palette-colorable k-graphs that avoid the forbidden family.

Load-bearing premise

The palette classification tools correctly characterize the existence of k-graphs satisfying any prescribed palette colorability constraints on their (k-2)-subsets.

What would settle it

A concrete family of k-graphs for which the supremum density of arbitrarily large F-free k-graphs that are uniformly d-dense on every (k-2)-subset differs from the maximum density permitted by the corresponding palette colorability constraints.

read the original abstract

For $k\ge 3$, the $(k-2)$-uniform Tur\'an density $\pi_{k-2}(F)$ of a $k$-graph $F$ is the supremum of $d$ for which there are arbitrarily large $F$-free $k$-graphs that are uniformly $d$-dense with respect to the $k$-vertex cliques of every $(k-2)$-graph on the same vertex set. We develop a \emph{palette framework} for this density. For every family $\mathcal F$ of $k$-graphs, we prove that $\pi_{k-2}(\mathcal F)$ equals the corresponding palette Tur\'an density. We further establish palette classification tools for the existence of $k$-graphs satisfying prescribed palette colorability constraints. Those together allow us to reduce exact density computations to a palette-homomorphism framework without relying on the hypergraph regularity method. As applications, for all $k\ge 3$ and $r\ge 2$, we establish the following values \[ \frac{r-1}{r},\quad \frac{(r-1)^2}{r^2},\quad \frac{r-1}{2r},\quad \frac{(k-1)^k}{k^k},\quad \frac{4(k-2)^{k-2}}{k^k},\quad \frac{4(k-2)^{k-2}}{3k^k} \] as $(k-2)$-uniform Tur\'an densities of single $k$-graphs. Finally, for every $k\ge3$, we show that there exist $k$-graphs $F_1,F_2$ such that \[ \pi_{k-2}(\{F_1,F_2\})< \min\{\pi_{k-2}(F_1),\pi_{k-2}(F_2)\}, \] which provides the first examples of \emph{non-principal} families for this density.

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 / 2 minor

Summary. The paper introduces a palette framework for the (k-2)-uniform Turán density π_{k-2}(F) of k-graphs F (k≥3). It proves that for any family F, π_{k-2}(F) equals the corresponding palette Turán density, develops palette classification tools for existence of k-graphs with prescribed colorability constraints, and reduces exact computations to a palette-homomorphism framework without regularity. Applications include establishing the exact values (r-1)/r, (r-1)^2/r^2, (r-1)/(2r), (k-1)^k/k^k, 4(k-2)^{k-2}/k^k, and 4(k-2)^{k-2}/(3k^k) as (k-2)-uniform Turán densities for single k-graphs (all k≥3, r≥2), plus the existence of k-graphs F1, F2 with π_{k-2}({F1,F2}) < min{π_{k-2}(F1), π_{k-2}(F2)}.

Significance. If the equivalence and classification tools hold, the work supplies a combinatorial reduction that bypasses the hypergraph regularity lemma for uniform Turán densities, yields the first explicit exact values for several infinite families of k-graphs, and provides the first examples of non-principal behavior. These are load-bearing advances for the area, as they convert density questions into homomorphism problems on palettes.

major comments (2)
  1. [§3] §3 (Palette classification tools): the claim that these tools decide existence of k-graphs satisfying arbitrary prescribed palette colorability constraints without hidden restrictions on vertex sets or palettes is load-bearing for the equivalence π_{k-2}(F) = palette density. A concrete verification is needed that the tools capture all admissible constructions for the listed densities, e.g., (k-1)^k/k^k and 4(k-2)^{k-2}/k^k, including the case of the non-principal pair F1,F2.
  2. [Theorem 4.2] Theorem 4.2 (non-principal families): the strict inequality π_{k-2}({F1,F2}) < min{π_{k-2}(F1),π_{k-2}(F2)} relies on the classification tools producing admissible F1,F2; the manuscript must exhibit explicit F1 and F2 together with the palette-homomorphism calculation that witnesses the gap.
minor comments (2)
  1. [§2] Notation for palette-homomorphism density should be introduced with a displayed equation in §2 before its first use in the main equivalence statement.
  2. [§4] The list of six explicit densities in the abstract and §4 would benefit from a table cross-referencing each value to the corresponding k-graph and the palette configuration used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below. We will revise the manuscript to incorporate explicit verifications and constructions as requested.

read point-by-point responses

  1. Referee: [§3] §3 (Palette classification tools): the claim that these tools decide existence of k-graphs satisfying arbitrary prescribed palette colorability constraints without hidden restrictions on vertex sets or palettes is load-bearing for the equivalence π_{k-2}(F) = palette density. A concrete verification is needed that the tools capture all admissible constructions for the listed densities, e.g., (k-1)^k/k^k and 4(k-2)^{k-2}/k^k, including the case of the non-principal pair F1,F2.

    Authors: We agree that explicit verification will strengthen the presentation. In the revised manuscript we will add a dedicated subsection to §3 containing concrete k-graph constructions for each listed density, including (k-1)^k/k^k and 4(k-2)^{k-2}/k^k. Each construction will be accompanied by a direct check that it satisfies the prescribed palette colorability constraints given by the classification tools, confirming that no hidden restrictions on vertex sets or palettes arise. The same subsection will treat the non-principal pair F1,F2 by exhibiting the specific palette constraints used to produce them. revision: yes

  2. Referee: [Theorem 4.2] Theorem 4.2 (non-principal families): the strict inequality π_{k-2}({F1,F2}) < min{π_{k-2}(F1),π_{k-2}(F2)} relies on the classification tools producing admissible F1,F2; the manuscript must exhibit explicit F1 and F2 together with the palette-homomorphism calculation that witnesses the gap.

    Authors: We will exhibit explicit F1 and F2 in the revised version of Theorem 4.2. The constructions will be given as concrete k-graphs whose forbidden subgraphs correspond to specific palette-homomorphism obstructions. We will include the full palette-homomorphism calculations that establish the strict inequality between the joint density and the individual densities, obtained by determining the maximum density of palettes that avoid homomorphisms to both F1 and F2. revision: yes

Circularity Check

0 steps flagged

No circularity: new palette framework reduces density to independent homomorphism quantity

full rationale

The paper introduces a palette framework and proves that π_{k-2}(F) equals the palette Turán density for arbitrary families F of k-graphs. Palette classification tools are developed internally to characterize existence under colorability constraints, enabling reduction to a palette-homomorphism framework without regularity methods or external self-citations. No derivation step equates a claimed prediction or density value to a fitted parameter or self-defined input by construction. The listed explicit densities and the non-principal family example follow directly from the combinatorial characterization rather than renaming or circular reduction. The argument is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the newly defined palette framework and the classification tools for colorability constraints; these are introduced in the paper rather than derived from prior results.

axioms (1)
  • standard math Standard axioms of set theory and finite combinatorics
    Used implicitly for the existence of finite hypergraphs and colorings.
invented entities (1)
  • Palette framework no independent evidence
    purpose: Models uniform density via color assignments to lower-dimensional cliques
    Newly developed in the paper to equate the uniform Turán density to a homomorphism density.

pith-pipeline@v0.9.0 · 5670 in / 1358 out tokens · 60137 ms · 2026-05-15T03:15:45.126868+00:00 · methodology

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Reference graph

Works this paper leans on

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