Uniform Tur\'an density -- palette classification

Ander Lamaison; Daniel Kr\'al'; Filip Ku\v{c}er\'ak; G\'abor Tardos

REVIEW 3 major objections 3 minor 1 cited by

Uniform Tur\'an density -- palette classification

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A finite no-homomorphism rule decides palette separation.

desk verdict Useful palette classification that strengthens the main construction tool for uniform Turán densities; sound modulo two fixable presentation gaps. read the letter →

arxiv 2505.17325 v2 pith:FLK74PQG submitted 2025-05-22 math.CO

Daniel Král' , Filip Kučerák , Ander Lamaison , Gábor Tardos This is my paper · ORCID

classification math.CO MSC 05C6505C3505D10

keywords 3-uniformhypergraphsuniformTurándensitypalettehomomorphismsymmetrizationhypergraphcolorabilityRamseytheoryregularitylemma

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 answers, for 3-uniform hypergraphs, a classification question about palette colorings: given palettes $P_1,\dots,P_r$ and $P_0$, when does a hypergraph exist that is colorable with every $P_i$ but not with $P_0$? The answer it proves is a finite, easily checked condition: such a hypergraph exists exactly when, for every $q$, there is no homomorphism from the palette $P_q\times\prod_{s\neq q}P_s^{(s)}$ to $P_0$ or to its inverse $\mathrm{inv}(P_0)$. This is relevant because a recent theorem identifies the uniform Turán density of a hypergraph with the supremum of the densities of palettes that fail to color it; the classification therefore turns the search for hypergraphs with prescribed uniform Turán density into a finite palette-comparison problem. As an illustration, the paper constructs a hypergraph whose uniform Turán density is $4/81$, a value not previously known to occur.

What carries the argument

The machinery centres on three palette operations and one transfer principle. A homomorphism between palettes is a colour map that sends every feasible triple to a feasible triple, so a homomorphism from $P$ to $P'$ forces every $P$-colorable ordered hypergraph to be $P'$-colorable. The inverse palette $\mathrm{inv}(P)$ reverses the order of triples, corresponding to reversing the vertex order. The symmetrization $P^{(s)}$ duplicates each colour and, for each feasible triple, adds the six triples obtained by permuting the vertex order and cloning colours whose left/right role flips; it makes colorability insensitive to which vertex order witnesses it. The multi-palette theorem forms the palette $P_q\times\prod_{s\ne q}P_s^{(s)}$ because, when all $P_s$ colour the same hypergraph, the $q$-th palette supplies the master order and every other palette can be used in symmetrized form. The proof also leans on regularity-lemma arguments and Ramsey-type results to locate monochromatic subsystems and to force the existence of the homomorphism $h$.

What would settle it

Enumerate all palettes with up to four colours and all small 3-uniform hypergraphs, and compare the theorem's homomorphism condition with the actual existence of a separating hypergraph; a single disagreement would refute the classification. Alternatively, exhibit three regular-looking vertex classes with pairwise density at least $2\varepsilon$ that contain no triangle, which would break the quoted embedding lemma.

Watch

Extended reading notes

Core claim

The central claim is Theorem 16 and its Corollary 17. In palette language, a palette is a finite set of colours together with a set of feasible ordered triples of colours, and an ordered hypergraph is $P$-colorable when its vertex pairs can be coloured so that every edge receives a feasible triple. The theorem states that a hypergraph $H$ that is $P_s$-colorable for every $s\in[r]$ but not $P_0$-colorable exists if and only if, for every $q\in[r]$, there is no homomorphism from the palette $P_q\times\prod_{s\in[r]\setminus\{q\}}P_s^{(s)}$ to $P_0$ or to $\mathrm{inv}(P_0)$, where $P^{(s)}$ is the symmetrization of $P$: two twin colours per original colour and, for each feasible triple, all six permutations with colours cloned when their left/right role is reversed. The proof has two directions: absence of the homomorphism is used, via regularity and Ramsey-type arguments, to build a separating hypergraph; presence of the homomorphism is used to show every $P_s$-colorable hypergraph must be $P_0$-colorable, because homomorphisms transfer colorability. The paper also proves the $r=1$ special case, where the condition simplifies to checking homomorphisms from $P$ to $P_0$ or to $\mathrm{inv}(P_0)$.

Load-bearing premise

The proof quotes without proof a regularity-embedding lemma saying that three large, well-mixed vertex sets with pairwise edge density at least a small threshold must contain a triangle of prescribed edge colours; the main characterization relies on that statement.

Editorial extensions

If this is right

The classification is decidable in finite terms: to test whether a separating hypergraph exists, one only needs to check finitely many homomorphisms between the given finite palettes.
When $r=1$, the criterion reduces to Theorem 13: a hypergraph colorable with $P$ but not with $P_0$ exists exactly when there is no homomorphism from $P$ to $P_0$ and no homomorphism from $P$ to $\mathrm{inv}(P_0)$.
The symmetrization factor is essential; the paper shows with the broken tetrahedron example that the plain product $P_1\times\cdots\times P_r$ would give a false criterion.
Given the palette characterization of uniform Turán density, the criterion yields hypergraphs with specified uniform Turán densities; the paper exhibits one with density exactly $4/81$.
The same necessary-and-sufficient condition extends, via disjoint unions, to separating a hypergraph from finitely many palettes at once, as stated in Corollary 17.

Reading between the lines

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

The criterion is algorithmic in character, so one could enumerate palettes up to a bounded number of colours and search for new attainable uniform Turán densities; the paper states the condition is easy to verify but does not develop this search procedure.
Because the proof uses only finite palettes and finite transfer arguments, a similar classification likely applies to other ordered colouring problems built from finite sets of feasible triples, not only to uniform Turán density.
The symmetrization operation may be relevant for $k$-uniform analogues if a palette characterization of uniform Turán density is later established in that setting; the paper itself is explicitly limited to 3-uniform hypergraphs.
The new multidimensional sign-pattern Ramsey theorem used in the proof may be useful independently for other problems requiring simultaneous control of several linear orders on a grid.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Palette classification preprint. The main theorem is real: a finite checkable criterion for the existence of a hypergraph colorable by every palette in one list and by none in another. That is exactly the tool you want for constructing lower-bound examples for uniform Turán density, and the 4/81 application produces a density value not previously realized. The r=1 case was already proved by King et al., and the paper says so; the multi-palette version with symmetrization is new and needs substantially more work.

The strengths are genuine. The proof of the central equivalence (Theorem 16 and its corollary) is long but coherent: regularity lemma preprocessing, a random embedding argument, and a Ramsey step that forces a homomorphism. The symmetrization operation is motivated and the conclusion section explains why the plain product would fail. Lemma 15's use of Fishburn–Graham lexicographic Ramsey theory is a good fit. No circularity, no fitted parameters; the 4/81 example is a genuine consequence of the characterization.

Two soft spots, both minor but worth fixing. Lemma 5, the triangle-embedding lemma for three regular pairs, is stated without proof. It is true and the proof is routine, but in a technical paper the authors should include it or give a precise citation. More interestingly, the stress-test note on Theorem 9 holds: the proof via Ramsey on 2r-subsets only gives the stated equality when the two r-tuples have no equal coordinates. The statement as written includes equal-coordinate pairs, and for those the union of coordinates can have fewer than 2r elements, so the argument does not go through. However, Lemma 14 only invokes the conclusion for triples with all coordinates distinct, so a weaker statement (require x_s≠y_s for every s) suffices and the main theorem stands. The authors should adjust Theorem 9's hypothesis to match what is actually used.

I'd send this to a serious referee. It is a paper for people who build palette constructions or study uniform Turán densities; for them it will be a standard reference. Recommendation: referee, conditional on the two fixes—proof or citation for Lemma 5, and a corrected Theorem 9 statement.

Referee Report

3 major / 3 minor

Summary. The paper studies the palette method for uniform Turán densities of 3-uniform hypergraphs. Building on Lamaison's theorem that the uniform Turán density equals the supremum of Lagrangians of non-coloring palettes, the authors give a necessary and sufficient condition for the existence of a hypergraph that is colorable with each of several given palettes P_1,...,P_r but not with another given palette P_0 (Theorem 16), and the corresponding multi-target version (Corollary 17). The condition is finite and checkable: it asks for the nonexistence of homomorphisms from certain product palettes involving the symmetrization operation to P_0 or its inverse. The paper also gives the single-palette case (Theorem 13) and demonstrates the utility of the criterion by constructing a 3-uniform hypergraph with uniform Turán density exactly 4/81, a value not previously known to occur.

Significance. If the amendments below are made, the paper constitutes a substantial contribution. The characterization of palette separation is a natural and useful completion of the palette program: it turns an existence question about hypergraphs into a finite homomorphism check. The introduction of the symmetrization operation is conceptually important, and the proof of the multi-palette theorem is an intricate piece of work using regularity, Ramsey theory, and Erdős–Szekeres arguments in what appears to be a correct combination once the stated gaps are repaired. The application yielding a hypergraph of uniform Turán density 4/81 is a concrete, falsifiable new result. The paper is honest about its reliance on Lamaison's theorem as an external benchmark, and the main characterization is proved independently of that theorem.

major comments (3)

[Section 3 (Theorem 9)] Theorem 9 is stated for all pairs x,y in J_1×...×J_r, including pairs with equal coordinates in some positions. The proof, however, colors (2r)-element subsets of [R] and defines the function C from such a subset. This only determines F(x,y) when the union of the coordinates of x and y has exactly 2r elements, i.e. when x_s≠y_s for every s. When some coordinates coincide, the union has fewer than 2r elements, and extending it to a 2r-subset can shift the positions of the relevant pairs; the value of C at vectors containing a zero is not forced by the Ramsey construction. Lemma 14 uses Theorem 9 only for triples i,i',i'' with all coordinate pairs distinct, so a weaker statement requiring x_s≠y_s for every s (with C defined on {−1,1}^r) suffices. Please restate Theorem 9 accordingly, or provide a genuinely different proof for the equal-coordinate case.
[Section 2 (Lemma 5) and its uses in Lemmas 11 and 14] Lemma 5 is stated as an uncolored triangle-existence statement: for three ε-regular pairs with density at least 2ε, there exist mutually adjacent vertices v_1,v_2,v_3. In the proofs of Lemma 11 and Lemma 14, however, the lemma is invoked to produce a triangle whose three edges have prescribed colors in the edge-colored complete graph, for example (x_{12},f_0(x_{12})), (x_{13},f_0(x_{13})), (x_{23},f_0(x_{23})) in Lemma 11. That is a colored triangle-embedding statement and does not follow from the uncolored Lemma 5 as formulated. The authors should either strengthen Lemma 5 to a lemma that simultaneously realizes a prescribed three-edge color pattern in three ε-regular pairs, or add such a lemma as a separate statement, and supply a proof or a precise reference. This lemma is load-bearing for the homomorphism constructions in both the single-palette and multi-palette cases.
[Sections 4 and 5 (Lemma 11 and Lemma 14)] The Ramsey parameters in the two lemmas are numerically inconsistent with the objects being colored. In Lemma 11 the number R is chosen using ℓ=|C_0||C|, but the auxiliary graph G' is edge-colored by mappings f:C→C_0, of which there are |C_0|^{|C|}. Similarly, in Lemma 14 the parameter ℓ in the application of Theorem 9 is written as |C_0|ℓ_C, whereas the functions F(i,i') take values in the set of all functions C_1×...×C_r→C_0, of cardinality |C_0|^{ℓ_C}. With the printed values R may be too small to guarantee the required monochromatic triangle or the required grid. Replace these products by the exponential counts |C_0|^{|C|} and |C_0|^{ℓ_C}, respectively.

minor comments (3)

[Section 2 (Lemma 5)] Once Lemma 5 is strengthened as suggested in the major comments, please supply the full argument or a specific reference; the current sentence 'The proof of the next lemma follows from standard regularity method arguments, and we omit it' is too terse for a lemma used twice in central homomorphism proofs.
[Section 6 (Lemma 20)] The displayed feasible triples for P_LM×P_{3T}^{(s)} and P_{3T}×P_{LM}^{(s)} appear to suppress the clone bars in the symmetrized coordinates. For example, the second triple of the first product should presumably have third entry (γ',\bar{ω}) rather than (γ',ω), since the underlying permutation requires the clone of ω. Please clarify the convention, because in the symmetrized palette the clone is a distinct color from the original.
[Section 6 (Theorem 21)] The sentence 'and so with any palette P with Lagrangian larger than 4/81' does not follow from the preceding statement about palettes with density larger than 4/81, since L(P)>4/81 does not imply d(P)>4/81. The upper bound is obtained instead by applying the density form of Theorem 2 (or equivalently the remark that density can replace Lagrangian there). Please rephrase this step to avoid the non-sequitur.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main palette-separation criterion is proved from first principles; the self-citations are to independent external results.

full rationale

The central characterization (Theorem 16 and Corollary 17) is self-contained. The 'only if' direction shows directly that a homomorphism from P_q × ∏_{s≠q} P_s^(s) to P_0 or inv(P_0) forces every hypergraph colorable by all P_s to be P_0-colorable, using the symmetrization construction and Proposition 3. The 'if' direction is the contrapositive: assuming no such homomorphism, Lemma 14 (proved via random coloring, the regularity lemma, Ramsey's theorem and Lemma 5) produces ordered hypergraphs H_q^+ and H_q^-, and Lemma 15 (a probabilistic construction with independent events) combines them into the desired hypergraph. None of these steps assumes the existence of the sought hypergraph, fits a parameter to the desired conclusion, or imports a uniqueness theorem. The definition of symmetrization is novel and justified in the proof, so it is not an ansatz smuggled in by citation. The paper does cite Lamaison's Theorem 2 [18] in the Section 6 application to convert palette colorability into uniform Turán density, and [19] in the conclusion; these are by the third author, but they cite independent external theorems and do not support the main classification. Two non-circular caveats are flagged: Lemma 5 is stated without proof ('The proof of the next lemma follows from standard regularity method arguments, and we omit it.'), and Theorem 9's proof only establishes the displayed equality when the coordinate pairs are distinct, so its statement overreaches for pairs with x_s = y_s; Lemma 14 only needs the distinct-coordinate case, so the main theorem is unaffected. These are correctness/exposition issues, not circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted; the paper is a proof-based contribution. It relies on standard combinatorial tools and on two external theorems: Lamaison's Theorem 2 and the Fishburn-Graham theorem. No new entities are postulated.

assumptions (7)

domain assumption Theorem 2 (Lamaison): the uniform Turán density of any 3-uniform hypergraph equals the supremum of Lagrangians of palettes it is not colorable with.
Invoked in the Introduction and in Theorem 21; proved in Lamaison [18], not in this paper.
standard math Lemma 4: Szemerédi Regularity Lemma for prepartitioned edge-colored graphs.
Stated in Section 2 without proof; used in Lemma 11 and Lemma 14.
standard math Lemma 5: triangle embedding in ε-regular triples with density at least 2ε.
Stated in Section 2, proof omitted; used in Lemma 11 and Lemma 14 to find vertices with prescribed edge colors.
standard math Theorem 7 (Erdős-Szekeres): any long sequence contains long increasing or decreasing subsequence.
Used in the proof of Lemma 12.
standard math Theorem 8 (Ramsey's Theorem, multicolor hypergraph version).
Used in Lemma 11 and in the proof of Theorem 9.
standard math Theorem 10 (Fishburn-Graham lexicographic Ramsey theorem).
Used in Lemma 15 to find subgrids with lexicographic order.
standard math Chernoff Bound for concentration of edge color counts.
Used in Lemma 11 and Lemma 14 to show typical colorings have nearly uniform color densities.

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Pith. "Pith review of Uniform Tur\'an density -- palette classification." pith.science (2026). https://pith.science/paper/FLK74PQG

@misc{pith2026250517325,
  author       = {Pith},
  title        = {Pith review of: Uniform Tur\'an density -- palette classification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FLK74PQG}},
  note         = {Machine review of arXiv:2505.17325}
}

abstract

In the 1980s, Erd\H{o}s and S\'os initiated the study of Tur\'an hypergraph problems with a uniformity condition on the distribution of edges, i.e., determining density thresholds for the existence of a hypergraph H in a host hypergraph with edges uniformly distributed. In particular, Erd\H{o}s and S\'os asked to determine the uniform Tur\'an densities of the hypergraphs $K_4^{(3)-}$ and $K_4^{(3)}$. After more than 30 years, the former was solved by Glebov, Kr\'al' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, R\"odl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter still remains open. In these two cases and several additional cases, the tight lower bounds are provided by a so-called palette construction. Lamaison [arXiv:2408.09643] has recently showed that the uniform Tur\'an density of a 3-uniform hypergraph H is equal to the supremum of the densities of palettes that H is not colorable with. We give a necessary and sufficient condition, which is easy to verify, on the existence of a 3-uniform hypergraph colorable by a set of palettes and not colorable by another given set of palettes. We also demonstrate how our result can be used to prove the existence of 3-uniform hypergraphs with specific values of the uniform Tur\'an density.

Figures

Figures reproduced from arXiv: 2505.17325 by the authors.

**Figure 2.** Visualization of a sought “subgrid” in the statement of Theorem 9 for [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** The feasible triples of the palette PLM. α β ω ω β ′ ω ′ ω ′ β ′′ γ ′′ [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: The feasible triples of the palette P4/81. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Finite palette endpoints and degree-square Tur\'an problems
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Proves exact degree-square Turán formulas for tournament palettes via auxiliary digraphs and majorization, yielding finite 3-graphs with uniform densities approaching 1/3.

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Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Buci´ c, J

M. Buci´ c, J. W. Cooper, D. Kr´ al’, S. Mohr and D. Munh´ a Correia:Uniform Tur´ an density of cycles, Transactions of the AMS376(2023), 4765–4809. 25

work page 2023

[2] [2]

Buci´ c, B

M. Buci´ c, B. Sudakov and T. Tran:Erd˝ os-Szekeres theorem for multidi- mensional arrays, Journal of the European Mathematical Society25(2023), 2927–2947

work page 2023

[3] [3]

Chen and B

A. Chen and B. Sch¨ ulke:Beyond the broken tetrahedron,preprint arXiv:2211.12747

arXiv

[4] [4]

Chung and L

F. Chung and L. Lu:An upper bound for the Tur´ an numbert 3(n,4), Journal of Combinatorial Theory Series A87(1999), 381–389

work page 1999

[5] [5]

Erd˝ os:On the combinatorial problems which I would most like to see solved, Combinatorica1(1981), 25–42

P. Erd˝ os:On the combinatorial problems which I would most like to see solved, Combinatorica1(1981), 25–42

work page 1981

[6] [6]

Erd˝ os:Problems and results on graphs and hypergraphs: similarities and differences, in: J

P. Erd˝ os:Problems and results on graphs and hypergraphs: similarities and differences, in: J. Neˇ setˇ ril and V. R¨ odl (eds.), Mathematics of Ramsey theory (1990), 223–233

work page 1990

[7] [7]

Erd˝ os and M

P. Erd˝ os and M. Simonovits:A limit theorem in graph theory, Studia Sci. Math. Hungar.1(1966), 51–57

work page 1966

[8] [8]

Erd˝ os and V

P. Erd˝ os and V. T. S´ os:On Ramsey-Tur´ an type theorems for hypergraphs, Combinatorica2(1982), 289–295

work page 1982