{"id":"1c3f7947-c32c-4713-8408-5e89560be9c8","arxiv_id":"2411.16218","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For each fixed k, the canonical Ramsey number ER(K^(k)_{t,...,t}) is at most t^{t^{k^2}} for large t, giving a single-exponential upper bound.","lead":"This paper proves upper bounds showing that canonical Ramsey numbers for partite hypergraphs grow at most single exponentially for every fixed uniformity. The result extends known bipartite bounds to hypergraphs and resolves a problem raised by Dobák and Mulrenin.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The k=2 proof in §3 fails its displayed inequalities: with δ0=δ1=2^{-6}t^{-3} and n=t^{3(t+1)}, both (3.1) and the second part of (3.2) evaluate to 2^{-8t}t^3<2t, so Proposition 2.1 cannot be applied. The main k≥3 result is unaffected.","rationale":"The reader's verdict is conditional because the k=2 constants fail (3.1)-(3.2). I agree with that identification. The central advance is the k≥3, and particularly k≥4, single-exponential upper bounds; those follow from Propositions 2.1-2.4 with the general parameter choice n=t^{t^{k^2}}, and a check of the exponent balance shows (3.1)-(3.5) hold for k≥3. The apparent 'box principle' in the j*=1 case is sound if read as producing a subset of size √(δ1 n) from a set of size δ1 n: if a color class of that size exists, take it; otherwise there are more than √(δ1 n) color classes, from which one can choose a rainbow subset. No other load-bearing gap emerged. The k=2 clause of Theorem 1.2 is not proved by the text as printed, but it is already known from the cited recent work, so the scientific claim of the paper is not false; the manuscript needs a correction or an attribution. Thus I would not change the conditional verdict.","tokens_in":8958,"tokens_out":20367,"duration_ms":175112,"concrete_test":"Write a short script or hand computation evaluating (3.1)-(3.5) for k=2,3,4 at t=10,20,50 with the constants stated in §3. For k=2, (3.1) and the second inequality of (3.2) will be false (≈2^{-8t}t^{3}), while for k=3 and k=4 all inequalities should hold. Then recompute the k=2 inequalities with the general-case choice δ0=δ1=1/(2^5 t^{2}), n=t^{t^{4}} to confirm they hold. This distinguishes the localized k=2 defect from the main result.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is the k=2 case of Theorem 1.2. The proof claims that for k=2 the same argument yields ER(K_{t,t}^{(2)}) ≤ t^{3(t+1)} with δ1=δ0=2^{-6}t^{-3}. This is false as printed. Inequality (3.1) for k=2 becomes (δ0/4)^{t} n = (2^{-8}t^{-3})^{t} t^{3t+3} = 2^{-8t}t^{3}, which is smaller than 2t for all sufficiently large t. Likewise, the second inequality in (3.2) is the same expression and fails. Since j*=0 and j*=1 are the only cases for graphs, neither can invoke Proposition 2.1, so the stated n is not derived. The failure is local: for the general choice δ0=δ1=c/t^{2} with n=t^{t^{4}} the same inequalities hold, so the single-exponential claim for k=2 survives in weakened form, and the sharper bound is already attributed to Gishboliner et al. and Dobák–Mulrenin. The k≥3 and k≥4 arguments balance; I find no analogous gap there. The paper should either correct the constants/case split or cite the known k=2 result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9230,"tokens_out":19276,"duration_ms":152638,"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":[{"comment":"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.","section":"§3, final paragraph (k=2 case)"},{"comment":"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.","section":"§3, inequalities (3.1)–(3.5)"}],"minor_comments":[{"comment":"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.","section":"§3, displayed order of constants"},{"comment":"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.","section":"§3, definition of m_j"},{"comment":"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'.","section":"§2.2, Proposition 2.3 statement"},{"comment":"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.","section":"§2.1, proof of Proposition 2.1"},{"comment":"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'.","section":"§3, k=3 paragraph"}],"recommendation":"major_revision","confidential_remarks":"The k=2 failure is the only substantive technical issue I found; the k≥3 part appears sound and the central claim is recoverable. The authors should be asked to correct the k=2 constants in Theorem 1.2 or explicitly defer to [8,3], and to supply the missing verification of (3.1)–(3.5). The manuscript is otherwise within the scope of a combinatorics journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe main new content is the k≥3 part of Theorem 1.2: for every fixed uniformity, the canonical partite Ramsey number is at most t^{t^{O(1)}} (with explicit exponents 30t^3 for k=3 and t^{k^2} for k≥4). That is a real advance and it resolves the problem Dobák and Mulrenin raised. The proof is a careful adaptation of their approach, with a genuinely useful unbalanced extremal proposition (Proposition 2.1) proved in full. The bounded-coloring machinery is standard but handled cleanly. The paper also does the right thing by crediting the recent k=2 bounds to Dobák–Mulrenin and Gishboliner–Milojević–Sudakov–Wigderson; the k≥3 bounds are not in those papers.\n\nWhere it gets soft: the k=2 case as printed is not justified. The text says that for n=t^{3(t+1)} and δ1=δ0=2^{-6}t^{-3} \"one can check\" the inequalities hold. They don't. For k=2, the left side of (3.1) and the second half of (3.2) become (δ0/4)^t n = 2^{-8t} t^3, which is far below 2t for large t. So Proposition 2.1 cannot be applied in either branch of the non-bounded case. This is a local error: the k≥3 cases use different parameter regimes and I have no reason to doubt their inequalities; the reader's check of the displayed thresholds found no analogous gap. Moreover, since the k=2 bound is already known and cited, the fix is trivial: either separate δ0 and δ1 (or choose n=t^{4t} with δ's around c/t^2), or simply state the k=2 result as a consequence of the cited work.\n\nThe paper is honest about its limitations: it notes the gap to the lower bound for k=3 and says the method cannot get below exponent k^2/2. The citation pattern looks fine.\n\nSo: this deserves a serious referee. The central claim for k≥3 is new and appears sound; the k=2 glitch is a corrigendum, not a disproof of the method. I'd recommend sending it out, with a request to fix the k=2 parameters or relegate that case to the cited results. A combinatorics seminar would get something out of it; I'd bring it to our reading group.","headline":"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.","tokens_in":9784,"tokens_out":3061,"would_cite":true,"duration_ms":25054,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D10","05C35","05C65"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Ramsey theory","canonical colourings","partite hypergraphs","canonical Ramsey numbers","Erdős–Rado numbers","bounded colourings","extremal hypergraph theory","rainbow subhypergraphs"],"falsifier":"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.","tokens_in":8714,"feed_emoji":"🎨","tokens_out":23180,"duration_ms":175032,"temperature":0.7,"pith_summary":"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.","feed_headline":"Single-exponential bounds proved for canonical Ramsey numbers","feed_subtitle":"For partite hypergraphs, canonical patterns appear within one exponential — not iterated exponentials.","key_machinery":"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).","core_discovery":"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 $O(k)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"The proof follows this approach and resolves the problem it raised; its sharp graph bounds are the comparison point for $k=2$.","marker":"[3]"},{"why":"Erdős' extremal theorem for hypergraphs, the basis of the unbalanced partite counting result in Proposition 2.1.","marker":"[4]"},{"why":"The Erdős–Rado canonical Ramsey theorem that the paper quantifies in the partite setting.","marker":"[7]"},{"why":"Concurrent sharp bounds for canonical Ramsey numbers of graphs against which the $k=2$ result is measured.","marker":"[8]"},{"why":"The Kövari–Sós–Turán theorem, the graph extremal bound that Proposition 2.1 extends to $k$-partite hypergraphs.","marker":"[9]"},{"why":"Earlier quantitative work on canonical Ramsey numbers; contributes the bounded-colouring observation behind Propositions 2.3 and 2.4.","marker":"[10]"}],"fun_headline_variants":["Canonical Ramsey numbers for partite hypergraphs are single-exponential","Partite hypergraphs: canonical Ramsey numbers grow only single-exponentially","New proof: canonical Ramsey for partite hypergraphs is single-exponential","Single-exponential bounds: canonical Ramsey numbers on partite hypergraphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Canonical Ramsey numbers for partite hypergraphs are single-exponential","Partite hypergraphs: canonical Ramsey numbers grow only single-exponentially","New proof: canonical Ramsey for partite hypergraphs is single-exponential","Single-exponential bounds: canonical Ramsey numbers on partite hypergraphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000434,"raw_usage":{"total_tokens":2172,"prompt_tokens":871,"completion_tokens":1301,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":1223}},"tokens_in":487,"tokens_out":1301,"duration_ms":11422,"temperature":1.0,"reasoning_tokens":1223,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:24:57.800058+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Sharp exponents for bipartite Erd\\H{o}s-Rado numbers","cited_arxiv_id":"2410.08982","evidence_quote":"The proof follows this approach and resolves the problem it raised; its sharp graph bounds are the comparison point for $k=2$."},{"cited_title":"Canonical Ramsey numbers of sparse graphs","cited_arxiv_id":"2410.08644","evidence_quote":"Concurrent sharp bounds for canonical Ramsey numbers of graphs against which the $k=2$ result is measured."},{"cited_title":"Lefmann and V","cited_arxiv_id":null,"evidence_quote":"Earlier quantitative work on canonical Ramsey numbers; contributes the bounded-colouring observation behind Propositions 2.3 and 2.4."}],"review_version":1}