{"id":"0729c96f-3d9c-4b09-8771-6205e92addcc","arxiv_id":"2507.05641","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every k≥4 and C>1 there is a linear k-uniform hypergraph H with off-diagonal Ramsey number r(H,K_n^{(k)}) at least the (k-2)-fold tower of 2^{(log n)^C}.","lead":"This paper proves that off-diagonal Ramsey numbers of linear k-uniform hypergraphs can grow as (k-2)-fold exponential towers, for every k at least 4. It generalizes a recent 3-uniform breakthrough and shows that sparse forbidden hypergraphs do not keep Ramsey numbers polynomial.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The k≥4 tower-height result rests entirely on the k=3 base case imported from a shared-author preprint; this is the main load-bearing risk.","rationale":"The paper's internal proof is coherent: Proposition 1.3 together with Lemma 3.3 and the stepping-up lemmas gives a valid induction step from uniformity k−1 to k, and the isolated-vertex and constant-C issues are minor and fixable. The reader's conditional verdict already identifies the dependence on [4, Theorem 1.4] as the weakest assumption. I agree that this is the single most load-bearing concern, because without a correct k=3 base the entire tower-height statement for k≥4 lacks a foundation. The concrete check of re-deriving [4, Theorem 1.4] would settle whether this concern lands. Since the reader already marked the verdict CONDITIONAL and my analysis does not reveal a further flaw requiring a change, the verdict should remain unchanged.","tokens_in":15205,"tokens_out":37870,"duration_ms":399355,"concrete_test":"Obtain the proof of [4, Theorem 1.4] (arXiv:2404.02021) and re-derive it independently, checking specifically: (i) the constructed 3-graph is linear, (ii) the lower bound r(H,K_n^{(3)}) ≥ 2^{(log n)^C} is proved for every C>1, and (iii) no step of [4] invokes the results of the present paper. If all three hold, the induction in Section 3 proves Theorem 1.2; if any fails, the theorem is unproven for k≥4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.2 is proved by induction on k, with Proposition 1.3 as the induction step and Theorem 1.1 (from [4, Theorem 1.4]) as the base. The claim for every k≥4 therefore depends on the correctness of the k=3 base case. The present paper contains no proof of that base, and [4] is a preprint sharing an author with this paper, so the tower-height conclusion for k≥4 is not independently established within the manuscript. If [4, Theorem 1.4] has an error or an unverified hypothesis, the entire tower-height result for all higher uniformities collapses. The rest of the induction is internally sound: the isolated-vertex condition in Theorem 3.1 is not a serious obstruction because isolated vertices can be deleted from H without changing the Ramsey number up to a constant, and the factor-of-2 loss in the clique size in Proposition 1.3 can be absorbed by starting the induction from Theorem 1.1 with a slightly larger C. But neither repair addresses the missing independent verification of the base case.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies off-diagonal Ramsey numbers r(H, K_n^{(k)}) for linear k-uniform hypergraphs H. The main theorem (Theorem 1.2) asserts that for every C>1 and every k≥3, there exists a linear k-graph H with r(H, K_n^{(k)}) ≥ twr_{k-2}(2^{(\\log n)^C}) for all sufficiently large n, nearly matching the Erdős–Rado upper bound. The proof introduces a binary-tree reformulation of the stepping-up construction (Section 2), defines a linear hypergraph with a strong transversal property (Lemma 3.3), and proves an induction step (Proposition 1.3) that turns a lower bound for a linear (k−1)-graph into an exponential lower bound for a linear k-graph. The induction is based on the k=3 case imported from Conlon et al. [4, Theorem 1.4]. The paper also notes a separation between r(H, K_n^{(k)}) and r(H, K_{n,\\ldots,n}^{(k)}), and discusses explicit examples such as the Fano plane and its 4-uniform analogue.","tokens_in":15429,"tokens_out":23701,"duration_ms":241453,"significance":"If the results hold, this is a significant advance: it extends the recent k=3 breakthrough of Conlon et al. to all uniformities, showing that linearity of H does not force polynomial growth in r(H, K_n^{(k)}), and it gives a tower-type separation from the complete k-partite target. The paper's own contribution—the binary-tree stepping-up framework, the dynamic-programming bounds on the auxiliary function f, and the randomized construction of a linear hypergraph with the transversal property (Lemma 3.7)—is carefully developed and appears internally consistent. The proofs of the main construction are detailed and, apart from the issues below, reproducible, including a union bound and FKG argument in Lemma 3.7. However, the tower-height statement for all k≥4 is conditional on the correctness of the k=3 base case from the unpublished shared-author preprint [4], which is not proved in this manuscript.","major_comments":[{"comment":"The proof of Theorem 1.2 for k≥4 rests entirely on Theorem 1.1, which is imported from [4, Theorem 1.4]. This is an unpublished preprint sharing an author with the present paper, and its correctness is not established within this manuscript. If [4, Theorem 1.4] has an error or an unverified hypothesis, the claimed tower-height result for every k≥4 collapses. The authors should either include a self-contained proof of the k=3 base case, or explicitly state Theorem 1.2 as conditional on the correctness of [4] (and, if appropriate, cite a published version once it appears). As written, the abstract and introduction present the result unconditionally, which is not justified by the evidence in this paper.","section":"Section 1, Theorem 1.2 and the proof of Proposition 1.3"},{"comment":"Theorem 3.1 requires the input (k−1)-graph H to have no isolated vertices, but Proposition 1.3 is stated for an arbitrary linear (k−1)-graph H. The proof of Proposition 1.3 applies Theorem 3.1 directly without explaining how to handle isolated vertices. This is fixable by deleting isolated vertices (which does not change r(H, K_n^{(k-1)}) up to the stated bound), but the reduction is not stated. Please add this argument or adjust the hypotheses of Proposition 1.3 so that the proof is complete.","section":"Section 3, Theorem 3.1 and Proposition 1.3"}],"minor_comments":[{"comment":"The abstract says \"for any constant C>0\" but Theorem 1.2 requires C>1. Please align the statement.","section":"Abstract"},{"comment":"The vertex set of the stepping-up is written as \"{0, . . . , 2N − 1}\", which appears to conflict with the earlier description of vertices as binary strings of length N (which would give 2^N vertices). If the intended set is {0, . . . , 2^N − 1}, the typesetting should be corrected; if the intended size is 2N, the construction does not match the classical stepping-up and the proof of Proposition 1.3 would need rechecking.","section":"Section 2, Definitions 2.5 and 2.6"},{"comment":"The sentence \"We will proof by induction\" should read \"We will prove by induction\".","section":"Section 2, proof of Lemma 2.9"},{"comment":"In the definition of the random s-graph, the text says \"on k vertices\" but it should be \"on n vertices\".","section":"Section 3.2, proof of Lemma 3.7"},{"comment":"When applying Lemma 3.6 to a set I of size m < s−1, the paper does not explicitly note that I can be extended to a set of size s−1. Please add this clarification, as Lemma 3.6 is stated only for |I| = s−1.","section":"Section 3.2, proof of Lemma 3.7"},{"comment":"The SageMath code included in the proof is nonstandard for a journal article; if kept, it should be moved to a footnote or appendix, and the text should summarize the output more explicitly (the code prints only the empty set and the whole ground set).","section":"Section 4, Proposition 4.1"}],"recommendation":"major_revision","confidential_remarks":"The main new contribution, Proposition 1.3 and its proof, is substantial and appears sound. The central concern is that Theorem 1.2 for k≥4 depends on the unpublished shared-author preprint [4]; the editors may wish to consider the journal's policy on such dependencies. The isolated-vertex gap in Proposition 1.3 is minor and easily repaired. Overall, the paper is well within the journal's scope and, after the dependency is clarified and the small gaps are fixed, would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: the paper proves a genuine new lower bound for off-diagonal Ramsey numbers of linear k-graphs for k≥4, showing they can grow as a tower of height k−2 with an exponent (log n)^C. That is nearly best possible by the Erdős–Rado upper bound. The k=3 case is imported from Conlon et al. [4], a preprint sharing an author here, so the tower-height statement for all k≥4 rises or falls with that base case. The contributions in this paper are the binary-structure reformulation of stepping-up and the construction of a linear H′ with a transversal property; both look sound.\n\nThe proof is internally consistent. The two gaps the reader flagged are minor: Proposition 1.3 applies Theorem 3.1 which requires no isolated vertices, but deleting isolated vertices from H does not affect the Ramsey number except by a constant; and the factor-2 loss in the clique size in Proposition 1.3 can be absorbed by starting with a slightly larger C in Theorem 1.1. Neither changes the qualitative result.\n\nThe soft spot is the dependence on [4]. This is not a hidden flaw—the authors state it plainly. But it is load-bearing: every k≥4 conclusion rests on the k=3 base from a shared-author preprint, and the present paper cannot independently establish the tower-height result. A referee would need to verify [4] or at least ensure it is accepted and correct. The field routinely builds on preprints, so I do not count this as a fatal objection, but it is the main thing to check.\n\nThere is also a small typo in Lemma 2.9 ('We will proof') and a stray 'k' in the proof of Lemma 3.7 where the random s-graph is said to be on 'k vertices' but should be n. These are inconsequential.\n\nWho is this for? People working on hypergraph Ramsey numbers and stepping-up constructions. The binary-structure framework and the dynamic-programming bounds on f(n1,n2,T) are worth reading in their own right. I would send this to a serious referee; it deserves review. My own verdict: accept the result as correct assuming [4] checks out, with revisions for the minor issues and a request to state the base-case dependency even more explicitly.","headline":"New lower bounds for off-diagonal Ramsey numbers of linear hypergraphs in uniformity k≥4, built on a shared-author preprint for the k=3 base; the new stepping-up framework is the real contribution.","tokens_in":15943,"tokens_out":2365,"would_cite":true,"duration_ms":25153,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D10","05C65","05D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every uniformity k≥3, some linear k-graph makes r(H,K_n^{(k)}) a tower of height k−2.","keywords":["off-diagonal Ramsey numbers","linear hypergraphs","hypergraph Ramsey numbers","stepping-up construction","tower function","binary structures","independence number","tower-height separation"],"falsifier":"Compute r(H,$K_n^{{(3)}}$) for the specific linear 3-graphs produced by [4, Theorem 1.4]: if for some C>1 and some such H the value is bounded by a polynomial in n for infinitely many n, the base case (and hence Theorem 1.2 for all k≥4) is false. Alternatively, rerun the code in Proposition 4.1 to verify the claimed partition obstruction showing that the seven-vertex projective-plane 3-graph is absent from the recursive construction, which grounds the explicit exponential example.","tokens_in":15019,"feed_emoji":"📈","tokens_out":6542,"duration_ms":61032,"temperature":0.7,"pith_summary":"This paper proves that off-diagonal Ramsey numbers $r(H,K_n^{(k)})$ for fixed linear $k$-uniform hypergraphs can be enormous: for every $k\\ge 3$ and every constant $C>1$, there exists a linear $k$-graph $H$ with $r(H,K_n^{(k)})\\ge \\mathrm{twr}_{k-2}(2^{(\\log n)^C})$. The height-$k-2$ tower is nearly the largest allowed by the classical stepping-down upper bound $r(H,K_n^{(k)})\\le \\mathrm{twr}_{k-1}(n^{O_H(1)})$, so linear hypergraphs are not as tame as the old folklore conjecture suggested. The proof is an induction on uniformity: a stepping-up construction takes a linear $(k-1)$-graph with large off-diagonal Ramsey number and produces a linear $k$-graph whose Ramsey number is exponential in it, with the recent $k=3$ result as the base case. If correct, the same $H$ has polynomial $r(H,K_{n,\\ldots,n}^{(k)})$, separating the clique and complete-$k$-partite targets dramatically.","feed_headline":"Linear k-graphs force Ramsey numbers to tower height k-2","feed_subtitle":"Stepping-up nearly matches the classical upper bound and refutes polynomial growth for linear hypergraphs in every uniformity.","key_machinery":"The argument runs through a reformulation of the classical stepping-up construction. Vertices are integers; the top splitting level $\\ell(S)$ and left/right subsets of a set $S$ organize any $k$-set into a binary structure $b(S)$, whose internal-node levels supply the $\\delta$-sequence that determines whether the $k$-set is an edge. The new machinery defines left/right stepping-ups of a $(k-1)$-graph $G$ using increasing/decreasing binary structures, plus edge families of prescribed binary-structure type $T$, and an auxiliary independence bound $f(n_1,n_2,\\mathcal T)$ bounding how large a set can be without unwanted structures. Lemma 2.9 gives a linear bound on $f$ when $\\mathcal T$ contains two-leaf types $T_{a,b}$, and Lemma 2.10 gives a polynomial bound by depth. The hard part, Theorem 3.1, constructs the target linear $k$-graph $H'$ from an ordered expansion $H^+$ of $H$ via a randomized oriented $s$-graph (Lemma 3.3): any dyadic partition, two-coloring, and ordering contains a monochromatic order-respecting transversal copy of $H^+$, and this copy is used to embed $H$ into $G$, forcing the stepped-up graph to be $H'$-free.","core_discovery":"The central claim, Theorem 1.2, is that for every constant $C>1$ and every uniformity $k\\ge 3$ there is a linear $k$-uniform hypergraph $H$ for which $r(H,K_n^{(k)})\\ge \\mathrm{twr}_{k-2}(2^{(\\log n)^C})$ for all sufficiently large $n$. The authors establish this by proving Proposition 1.3, a stepping-up lemma: given a linear $(k-1)$-graph $H$, they construct a linear $k$-graph $H'$ with $r(H',K_{2n+2k}^{(k)})>2^{r(H,K_n^{(k-1)})-1}$, so one exponential step lifts the bound from uniformity $k-1$ to $k$. Starting from the $k=3$ theorem of the recent preprint [4], the induction gives tower height $k-2$. The construction avoids the polynomial upper bound for iterated $k$-partite hypergraphs, and it implies that $r(H,K_n^{(k)})$ and $r(H,K_{n,\\ldots,n}^{(k)})$ can be respectively tower-height and polynomial for the same linear $H$.","pith_inferences":["One could try to push the base case: replacing the $k=3$ construction's $2^{(\\log n)^C}$ with $2^{n^c}$ is the bottleneck; the induction in Proposition 1.3 is a clean exponential amplifier that would then yield height $k-1$ towers.","The dyadic-partition machinery is robust enough that Lemma 3.3 should hold for linear $s$-graphs of any fixed Berge girth, so the same stepping-up might produce linear $k$-graphs with large girth and tower Ramsey numbers; the paper notes the girth version in passing.","The separation between $r(H,K_n^{(k)})$ and $r(H,K_{n,\\ldots,n}^{(k)})$ suggests that the iterated-$k$-partite conjecture of [5] cannot be rescued by any linearity assumption; linearity alone does not force polynomial growth.","A concrete open target is the seven-vertex projective-plane $3$-graph: if one could prove super-polynomial $r(H,K_n^{(3)})$ for it, it would be the smallest linear hypergraph witnessing the failure of polynomial growth, and the present methods do not yet reach it."],"forward_implications":["For every $k\\ge 3$ there is a linear $k$-graph $H$ whose off-diagonal Ramsey number $r(H,K_n^{(k)})$ grows like a tower of height $k-2$, nearly matching the classical upper bound (1.1) of height $k-1$.","The same $H$ satisfies $r(H,K_{n,\\ldots,n}^{(k)})\\le n^{O_H(1)}$, so the Ramsey number against a single clique can be dramatically larger than against the complete $k$-partite hypergraph.","The folklore conjecture that every linear $3$-graph has polynomial $r(H,K_n^{(3)})$ is false, and this paper extends that failure to all uniformities.","An improved base case with $r(H,K_n^{(3)})\\ge 2^{n^c}$ would, by the same induction, give a linear $k$-graph with $r(H,K_n^{(k)})\\ge \\mathrm{twr}_{k-1}(n^c)$, matching the upper bound up to constants.","The fully explicit linear $4$-graph given by the lines of PG(2,3) yields an exponential lower bound and can seed an explicit tower construction of height $k-2$."],"supporting_citations":[{"why":"Supplies the k=3 base case (Theorem 1.1) that starts the induction; the present paper does not reprove it.","marker":"[4]"},{"why":"The classical stepping-up construction that is reformulated here in binary-structure language and generalized.","marker":"[10]"},{"why":"The stepping-down upper bound (1.1) that Theorem 1.2 nearly matches.","marker":"[11]"},{"why":"States the iterated-k-partite conjecture and the folklore context that motivates the class of linear hypergraphs.","marker":"[5]"},{"why":"Gives the supersaturation argument showing r(H,K_{n,...,n}^{(k)}) is polynomial for linear H, used to state the separation.","marker":"[14]"}],"fun_headline_variants":["Linear hypergraphs push Ramsey numbers to tower height k-2","Stepping up: linear k-graphs give tower Ramsey bounds","Ramsey towers for linear hypergraphs of any uniformity","Polynomial growth refuted: Ramsey numbers tower for linear k-graphs","Off-diagonal Ramsey: linear hypergraphs reach tower height k-2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The tower-height result for k≥4 rests entirely on the k=3 base case from the recent preprint [4]; if that base case is wrong, the induction has no starting point.","fun_headline_variants_meta":{"raw":{"variants":["Linear hypergraphs push Ramsey numbers to tower height k-2","Stepping up: linear k-graphs give tower Ramsey bounds","Ramsey towers for linear hypergraphs of any uniformity","Polynomial growth refuted: Ramsey numbers tower for linear k-graphs","Off-diagonal Ramsey: linear hypergraphs reach tower height k-2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000593,"raw_usage":{"total_tokens":2772,"prompt_tokens":935,"completion_tokens":1837,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":1747}},"tokens_in":551,"tokens_out":1837,"duration_ms":14230,"temperature":1.0,"reasoning_tokens":1747,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:23:59.255950+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute r(H,$K_n^{{(3)}}$) for the specific linear 3-graphs produced by [4, Theorem 1.4]: if for some C>1 and some such H the value is bounded by a polynomial in n for infinitely many n, the base case (and hence Theorem 1.2 for all k≥4) is false. Alternatively, rerun the code in Proposition 4.1 to verify the claimed partition obstruction showing that the seven-vertex projective-plane 3-graph is absent from the recursive construction, which grounds the explicit exponential example.","supporting_citations":[{"cited_title":"On off-diagonal hypergraph Ramsey numbers","cited_arxiv_id":"2404.02021","evidence_quote":"Supplies the k=3 base case (Theorem 1.1) that starts the induction; the present paper does not reprove it."},{"cited_title":"Erdős, A","cited_arxiv_id":null,"evidence_quote":"The classical stepping-up construction that is reformulated here in binary-structure language and generalized."},{"cited_title":"Erdős and R","cited_arxiv_id":null,"evidence_quote":"The stepping-down upper bound (1.1) that Theorem 1.2 nearly matches."},{"cited_title":"Fox and X","cited_arxiv_id":null,"evidence_quote":"Gives the supersaturation argument showing r(H,K_{n,...,n}^{(k)}) is polynomial for linear H, used to state the separation."}],"review_version":1}