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Canonical Ramsey numbers for partite hypergraphs

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Pith's one-line read The paper proves canonical Ramsey numbers for partite hypergraphs grow single-exponentially: $\mathrm{ER}(K^{(2)}_{t,t}) \leq t^{3(t+1)}$, $\mathrm{ER}(K^{(3)}_{t,t,t}) \leq t^{30t^3}$, and $\mathrm{ER}(K^{(k)}_{t,\ldots,t}) \leq…

desk verdict Genuinely new single-exponential upper bounds for canonical partite hypergraph Ramsey numbers in uniformity at least 3, with a local but easily fixed error in the printed k=2 constants. read the letter →

arxiv 2411.16218 v2 pith:WYUN3BA7 submitted 2024-11-25 math.CO

Matías Azócar Carvajal , Giovanne Santos , Mathias Schacht This is my paper · ORCID
classification math.CO MSC 05D1005C3505C65
keywords RamseytheorycanonicalcolouringspartitehypergraphsnumbersErdős–Radoboundedextremalhypergraphrainbowsubhypergraphs
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

This paper sets out to show that canonical Ramsey numbers for $k$-partite $k$-uniform hypergraphs grow only single-exponentially for each fixed uniformity $k$ — the same qualitative growth as ordinary partite Ramsey numbers, and vastly smaller than the iterated-exponential growth of canonical Ramsey numbers for complete (non-partite) hypergraphs. Concretely, Theorem 1.2 claims $\mathrm{ER}(K^{(2)}_{t,t}) \leq t^{3(t+1)}$, $\mathrm{ER}(K^{(3)}_{t,t,t}) \leq t^{30t^3}$, and $\mathrm{ER}(K^{(k)}_{t,\ldots,t}) \leq t^{t^{k^2}}$ for every $k \geq 4$ and sufficiently large $t$, complementing the lower bound $t^{(1-o(1))t^{k-1}}$ from random colourings with $t^k-1$ colours. If the theorem is right, it resolves a problem raised by Dobák and Mulrenin and, for $k=2$, produces an upper bound optimal up to the factor 3 in the exponent. The reader should care because the mechanism is structural: for partite hypergraphs the extremal problem is degenerate, and the canonical theory is shown to track that degeneracy, growing at the same exponential rate as the underlying Kövari–Sós–Turán-type extremal bounds.

What carries the argument

The carrying mechanism is the boundedness classification of colourings (Definition 2.2): for a set $J$ of vertex classes, a colouring is $(\delta,J)$-bounded if all but a $\delta$-fraction of the $|J|$-tuples in $V_J$ are each extended by at most $\delta\,|V_{[k]\setminus J}|$ edges of any single colour. The proof pivots on the minimal index $j^*$ at which the colouring fails to be bounded. If there is no such index, Proposition 2.3, a random-sampling lemma, produces a rainbow copy of $K^{(k)}_{t,\ldots,t}$ directly; the proposition formalises the known observation that bounded colourings contain large rainbow subhypergraphs. If $j^*$ exists, the witnessing tuples define a dense $j^*$-partite hypergraph, and an unbalanced variant of Erdős' extremal theorem (Proposition 2.1, itself a partite generalisation of the Kövari–Sós–Turán theorem) is applied twice — first to find a rainbow subhypergraph of controlled density, then to its $k$-uniform extension to force a $J^*$-canonical complete subhypergraph $K^{(k)}_{t,\ldots,t}$. Every application of Propositions 2.1–2.4 is justified by manually chosen constants $\delta_j$ and $m_j$ satisfying the inequalities (3.1)–(3.5).

What would settle it

Substituting the printed $k=2$ parameters $\delta_1=2^{-6}t^{-3}$ and $n=t^{3(t+1)}$ into the second inequality of (3.2) gives the requirement $(\delta_1/4)^t n = 2^{-8t}t^3 \geq 2t$, i.e. $2^{-8t}t^2 \geq 2$, which fails for every large $t$; this one calculation settles whether Theorem 1.2's $k=2$ bound is proved by the argument as printed.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the Erdős–Rado canonical Ramsey numbers for partite hypergraphs are controlled by the extremal (ordinary Ramsey) behaviour of the same hypergraphs. The proof classifies every edge-colouring of the complete $k$-partite $k$-uniform hypergraph with vertex classes of size $n$ into two regimes: if the colouring is $\delta$-bounded — meaning few colours proliferate on the edges extending any fixed tuple of vertices — a random-sampling argument (Proposition 2.3) yields a rainbow copy of $K^{(k)}_{t,\ldots,t}$; if some minimal index $j^*$ witnesses unboundedness, those offending tuples carry many edges of one colour and form a dense partite hypergraph, to which an unbalanced partite version of Erdős' hypergraph extremal theorem (Proposition 2.1) applies and forces a monochromatic, $\{1\}$-canonical, or $J^*$-canonical copy of $K^{(k)}_{t,\ldots,t}$. The resulting bounds are $t^{3(t+1)}$ for $k=2$, $t^{30t^3}$ for $k=3$, and $t^{t^{k^2}}$ for $k \geq 4$, all for sufficiently large $t$, with the $k=2$ bound optimal up to the factor 3 in the exponent. The paper leaves open whether the cubic exponent for $k=3$ can be reduced to quadratic, and whether the exponent $k^2$ in the general bound can be improved to $o(k^2)$ or even $O(k)$.

Load-bearing premise

The load-bearing premise is that the handpicked constants $\delta_j$ and $m_j$ satisfy the inequalities (3.1)–(3.5) for every uniformity $k$; as printed, the $k=2$ choice $\delta_0=\delta_1=2^{-6}t^{-3}$ with $n=t^{3(t+1)}$ fails the second inequality of (3.2), since $(\delta_1/4)^t n = 2^{-8t}t^3$ is smaller than $2t$ for large $t$, so the $k=2$ bound is not derived by the proof as written.

Editorial extensions

If this is right

  • For every fixed uniformity $k$, $\mathrm{ER}(K^{(k)}_{t,\ldots,t})$ is bounded by a single exponential in a polynomial of $t$, so canonical and ordinary partite Ramsey numbers have the same qualitative growth.
  • For $k=2$ the upper bound $t^{3(t+1)}$ is optimal up to the factor 3 in the exponent, matching the random-colouring lower bound and the independent graph bounds obtained concurrently by Gishboliner, Milojević, Sudakov, and Wigderson and by Dobák and Mulrenin.
  • For $k=3$ the theorem leaves the exponent $30t^3$ open to improvement, and the authors propose deciding whether a quadratic exponent is attainable.
  • For $k \geq 4$ the argument, as the authors state, cannot lower the exponent below $k^2/2$; improving $k^2$ to $o(k^2)$ or even $O(k)$ remains open.

Reading between the lines

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

  • I would expect the boundedness classification to be portable: in any hypergraph setting whose ordinary extremal problem is degenerate and well understood, the same two-regime argument should convert extremal bounds into canonical Ramsey bounds of the same growth type.
  • The gap between the lower bound $t^{(1-o(1))t^{k-1}}$ and the upper bound $t^{t^{k^2}}$ suggests the true exponent is far below $k^2$; the authors' own remark that the method cannot reach a factor below $1/2$ indicates the $k^2$ may be an artifact of the proof.
  • The $k=2$ parameter failure looks repairable without changing the theorem: the failing inequality misses by an exponential factor, so a rebalanced choice of the constants $\delta_0,\delta_1$ should satisfy (3.1)–(3.5) while preserving the claimed bound $t^{3(t+1)}$.
  • A natural next test of the method is the 3-uniform case, where replacing $30t^3$ by a quadratic exponent would narrow the gap to the lower bound and reveal whether the classification step or the extremal count is what limits the exponent.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper establishes single-exponential upper bounds for the canonical Ramsey numbers of complete k-partite k-uniform hypergraphs. Theorem 1.2 states ER(K^{(2)}_{t,t}) ≤ t^{3(t+1)}, ER(K^{(3)}_{t,t,t}) ≤ t^{30t^3}, and ER(K^{(k)}_{t,...,t}) ≤ t^{t^{k^2}} for k≥4 and large t. The proof combines an unbalanced Kővári–Sós–Turán-type counting lemma (Proposition 2.1), two lemmas on δ-bounded colourings (Propositions 2.3 and 2.4), and a case analysis on the minimal index j for which a colouring is not (δ_j,j)-bounded. For k≥3 the parameter choices are claimed to satisfy the displayed inequalities (3.1)–(3.5); for k=2 the sharper bound in the theorem is not supported by those inequalities as printed.

Significance. If the proof is completed, this resolves a problem of Dobák and Mulrenin and shows that partite canonical Ramsey numbers grow only singly exponentially for each fixed uniformity, in contrast with the full Erdős–Rado numbers. The paper is largely self-contained: Proposition 2.1 is proved in full, and the bounded-colouring propositions are proved. The k=3 and k≥4 arguments appear coherent, and the displayed inequalities hold with the stated large-t choices. The failure of the k=2 constant is local and repairable: the general argument with n=t^{t^4} and δ0=δ1=2^{-5}t^{-2} already gives a single-exponential bound for graphs, and the sharper t^{3(t+1)} bound is attributed to concurrent work by Gishboliner et al. and Dobák–Mulrenin. The central novel claim for k≥3 is not affected by this flaw.

major comments (2)
  1. [§3, final paragraph (k=2 case)] The claimed k=2 bound is not justified. With n=t^{3(t+1)} and δ1=δ0=2^{-6}t^{-3}, inequality (3.1) evaluates to (δ0/4)^t n = (2^{-8}t^{-3})^t t^{3t+3} = 2^{-8t}t^3, which is less than 2t for all sufficiently large t. The second inequality in (3.2) is the same expression, so Proposition 2.1 cannot be applied in either the j*=0 or the j*=1 case. Consequently the assertion ER(K^{(2)}_{t,t}) ≤ t^{3(t+1)} in Theorem 1.2 is unproved as printed. Please either correct the parameter choice (the general constants δ0=δ1=2^{-5}t^{-2}, n=t^{t^4} satisfy the relevant inequalities and give a weaker single-exponential bound), or cite the known k=2 results [8,3] and state the k=2 part of Theorem 1.2 accordingly.
  2. [§3, inequalities (3.1)–(3.5)] The proof of Theorem 1.2 rests entirely on the parameter inequalities (3.1)–(3.5), but the verification is only asserted ('one can check', 'mainly based on the facts that...'). Because the k=2 check is demonstrably wrong, the remaining cases need an explicit, complete verification of every displayed inequality with exact or sufficiently explicit thresholds for t. This is necessary for the theorem to be certified as stated; a sentence or short appendix would suffice.
minor comments (5)
  1. [§3, displayed order of constants] The chain '2 ≤ k < 1/c < t < 1/δ_{k−1} < m_{k−1} < ... < 1/δ_2 < m_2 < 1/δ_1 = 1/δ_0 < n' includes m_{k−1} and 1/δ_{k−1}, but for k=2 the parameter m_1 is not defined by the preceding display; the chain should be stated for k≥3 or adjusted for the graph case.
  2. [§3, definition of m_j] The displayed definition of m_j is ambiguous: the exponent appears as 't_k' and the equality with (t^k/c)^{(2k)^{k-1-j}t^k(k-j)} does not match the later k=3 choices (e.g., δ2=2^{-7}t^{-3} and m2=t^{7t}). Please clarify the intended exponent and state whether the general formula is meant for k≥4, with the k=3 values treated separately.
  3. [§2.2, Proposition 2.3 statement] The sentence 'let ϕ : E(K^{(k)}_{V_1,...,V_k}) → N be a δ-bounded colouring with of the complete k-partite...' contains a grammatical slip ('with of'); it should read 'of the complete k-partite hypergraph'.
  4. [§2.1, proof of Proposition 2.1] The phrase 'Jensen's inequality with weights 1/|V_k| for every v ∈ V*_k' is not literally correct, since those weights do not sum to 1. The subsequent inequality is valid after using uniform weights on V*_k and |V*_k|^{1−q} ≥ |V_k|^{1−q}, but the sentence should be rephrased.
  5. [§3, k=3 paragraph] The k=3 values δ2=1/(2^7 t^3), m2=t^{7t}, and δ1=δ0=1/(2^{10} t^{29t}) are stated without derivation; a short verification of (3.1)–(3.5) for these values would help the reader and reduce reliance on 'one can check'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is self-contained and derives the upper bound from independently proved extremal and bounded-coloring propositions.

full rationale

The paper's derivation chain is self-contained. The main tool, Proposition 2.1, is an unbalanced variant of the Kővári–Sós–Turán/Erdős hypergraph extremal result and is proved in the paper by induction from first principles. Propositions 2.3 and 2.4 are also proved directly via random subsets, Markov's inequality, and Chebyshev's inequality. The proof of Theorem 1.2 then proceeds by a case analysis on the minimal index j* where the given coloring fails to be δ-bounded: the j*=0 case invokes Proposition 2.1 to find a monochromatic copy, the j*=1 case invokes Proposition 2.1 again to find either a monochromatic or a {1}-canonical copy, and the j*≥2 case uses Proposition 2.4 to obtain a rainbow core and Proposition 2.1 to extend it to a J*-canonical copy. In each application, the hypotheses (3.1)–(3.5) are stated as inequalities that the chosen constants must satisfy, and the proof does not assume the target bound ER(K^(k)_{t,...,t}) ≤ t^{t^{k^2}} as an input. The citations to Dobák–Mulrenin and Gishboliner et al. for the k=2 case are contextual and the proof still attempts an independent derivation. The only substantive issue found is that the printed k=2 parameter choice δ0=δ1=2^{-6}t^{-3} and n=t^{3(t+1)} appears to make (3.1) and the second part of (3.2) fail, so the k=2 case as written may be a correctness gap rather than a valid derivation; however, that is an arithmetic/short-step issue, not circularity. No fitted parameter is relabeled as a prediction, no theorem is defined in terms of itself, and no load-bearing uniqueness claim rests on the authors' own prior work. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 2 assumptions · 0 invented entities

The central claim rests on standard probabilistic inequalities, an extremal hypergraph bound proven inside the paper, and the authors' hand-chosen parameter hierarchies. No new particles, forces, or external objects are introduced. The main unproved input is the assertion that (3.1)-(3.5) hold for all k; it holds for the novel cases but fails for the printed k=2 constants.

free parameters (2)
  • c = 1/2^{2k+1} = 1/2^{2k+1}
    A hand-chosen normalization constant used in all boundedness thresholds δ_j and in the proof of the main theorem.
  • δ_j and m_j = δ_j=(c/t^k)^{(2k t^k)^{k-1-j}} for j≥2, δ0=δ1=(c/t^k)^{(2k t^k)^{k-2}}, m_j=δ_j^{-t^k}
    Manually chosen proof parameters designed to make inequalities (3.1)-(3.5) hold and to balance the applications of Propositions 2.1-2.4. They are not empirical constants.
assumptions (2)
  • standard math Markov, Chebyshev, and Jensen inequalities are valid and applicable.
    Used without proof in Propositions 2.3 and 2.4 for concentration estimates and in Proposition 2.1 for the inductive double counting.
  • ad hoc to paper The chosen parameters satisfy the displayed inequalities (3.1)-(3.5) for sufficiently large t.
    This is asserted at the end of Section 3 and is load-bearing for the case analysis. The k=2 instance appears false as printed, which is exactly where the proof loses its justification.

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Pith. "Pith review of Canonical Ramsey numbers for partite hypergraphs." pith.science (2026). https://pith.science/paper/WYUN3BA7

@misc{pith2026241116218,
  author       = {Pith},
  title        = {Pith review of: Canonical Ramsey numbers for partite hypergraphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WYUN3BA7}},
  note         = {Machine review of arXiv:2411.16218}
}
read the original abstract

We show that canonical Ramsey numbers for partite hypergraphs grow single exponentially for any fixed uniformity.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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