{"id":"dd44c542-b484-46ac-96df-022984cfe356","arxiv_id":"2411.15649","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper establishes r(3;n) ≤ R(P_{n+2}, J_n) ≤ 4^n · r(3;n), connecting a central open Ramsey problem to a hypergraph monotone path problem.","lead":"This paper proves that the multicolor Ramsey number for triangles, r(3;n), is within a factor of 4^n of the Ramsey number of certain ordered 3-uniform monotone paths with 'jumps'. As a result, the famous question of whether r(3;n) is exponential is equivalent to whether that hypergraph Ramsey number is exponential.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The upper bound in Theorem 1.3 rests on an unproved and ill-specified assertion that β(u,v) > n forces a blue I_n, with the definitions of β and I_n inconsistent as written.","rationale":"The main claim of the paper is Theorem 1.3, which asserts a two-sided inequality linking r(3;n) and R(P_{n+2}, J_n). The lower bound in Section 2 is a standard reduction and appears sound: from an n-coloring of pairs avoiding monochromatic triangles, the derived red/blue coloring of triples avoids red P_{n+2} and blue J_n. The upper bound is where the argument is fragile. Its entire mechanism is to define α and β and to use the dichotomy α(x,y) ≥ α(y,z) ⇒ (x,y,z) blue to convert long β-chains into blue copies of I_n. The paper asserts this conversion without proof, and the surrounding definitions are not rigorous: the β condition's indexing is ambiguous at the boundary, and I_n as defined does not exist on [2n+1] because some specified triples fall outside the vertex set. These are not stylistic complaints; they are exactly the steps needed to justify the bound β ≤ n, which in turn is necessary for the counting argument that yields the factor 4^n. Without a complete proof of the 'β > n forces blue I_n' implication, the upper bound is unsupported. I also note the text says 'does not contain a red P_n' where the theorem requires P_{n+2}; this appears to be a typo, but it is another indication that the proof as written needs careful revision. The reader's verdict of CONDITIONAL is appropriate: the result may be true and the proof strategy plausible, but the manuscript as submitted does not fully demonstrate the central claim. No adjustment to the verdict is needed.","tokens_in":5349,"tokens_out":17270,"duration_ms":128317,"concrete_test":"Independently reconstruct the missing lemma: given a sequence v1 < ... < v_{2ℓ-1} with ℓ > n satisfying the β condition, explicitly identify the vertices of a copy of I_n and verify that each edge of I_n is blue using the rule α(x,y) ≥ α(y,z) ⇒ (x,y,z) blue. First repair the definition of I_n (for example, set the vertex set to [2n+2] and restrict the special triples to the appropriate range) and disambiguate the β condition by specifying the exact range of i. If the construction can be completed for all n, the upper bound is saved; if the verification fails for some n, Theorem 1.3 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central upper-bound argument in Section 2 depends entirely on the assertion 'Using this fact we can check that β(u,v) > n would force a blue I_n inside K_N^{(3)}' (right after the definition of β). This is the only step that yields the bound β(u,v) ≤ n, which is needed for D(v) ⊂ [n]^2 and the ≤4^n counting of downward-closed sets. The paper does not provide the construction: it does not explain how a witnessing sequence v1 < ... < v_{2ℓ-1} with ℓ > n is mapped to a copy of I_n, nor how each required edge of I_n is shown to be blue using α(x,y) ≥ α(y,z) ⇒ (x,y,z) blue. Moreover, the definitions involved are not well-formed as written. The definition of β has an indexing ambiguity: the condition 'for all possible i' references vertices v_{2i+2} and v_{2i+3} beyond the stated sequence length v1 < ... < v_{2ℓ-1} for i near ℓ. The specific hypergraph I_n is declared on vertex set [2n+1] but lists edges (2i,2i+1,2i+3) and (2i-1,2i+1,2i+3) which for i=n contain the vertex 2n+3 > 2n+1, and condition (1) for the last jump 2n requires the nonexistent vertex 2n+2. These are not mere typos in the text: they make the asserted check impossible to verify. If the check cannot be completed or the objects are corrected inconsistently, the upper bound R(P_{n+2}, J_n) ≤ 4^n·r(3;n) does not follow, and the equivalence with r(3;n) exponential is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a family J_n of ordered 3-uniform hypergraphs called monotone paths with n jumps and claims Theorem 1.3: r(3;n) ≤ R(P_{n+2}, J_n) ≤ 4^n · r(3;n). The lower bound is proved by translating an n-coloring of pairs avoiding monochromatic triangles into a red/blue coloring of K_N^{(3)} with no red P_{n+2} and no blue member of J_n. The upper bound uses auxiliary functions α and β, a counting argument over downward-closed subsets of [n]^2, and a pigeonhole step to bound N by 4^n · r(3;n). The paper concludes that the exponentiality of r(3;n) is equivalent to that of R(P_{n+2}, J_n).","tokens_in":5762,"tokens_out":22499,"duration_ms":175911,"significance":"If Theorem 1.3 is correct, it provides a new and potentially fruitful translation of a famous 100-year-old problem of Erdős into a hypergraph Ramsey question. The lower-bound construction is elegant and elementary, and the upper-bound counting via downward-closed sets is a nice application of known ideas of Moshkovitz–Shapira and Chvátal–Komlós. The paper is concise and mostly well written. However, the proof as written contains substantial gaps in both the lower and upper bound arguments, and the claimed equivalence is not established without substantial repair.","major_comments":[{"comment":"The claim “It is easy to check that H' ∈ J_{n-1}” is false in general. For example, let H be the ordered 3-uniform hypergraph on vertex set [7] whose edge set consists of all consecutive triples together with (1,2,4), (2,4,5), (3,4,6), (4,6,7), and (2,4,6). Taking J = {3,5}, one checks that H satisfies conditions (0)–(2) and hence H ∈ J_2. With w = 5 the largest jump, the induced subhypergraph H' on vertices {1,2,3,4} has only the candidate jump 3, but the edge (2,4,5) required by condition (1) for that jump is absent because vertex 5 is not in H'. Thus H' ∉ J_1, and the induction hypothesis cannot be applied. This invalidates the proof of the lower bound r(3;n) ≤ R(P_{n+2}, J_n).","section":"Section 2, lower bound"},{"comment":"The hypergraph I_n is declared to be a member of J_n on vertex set [2n+1], but as written it is not well defined. The edge list contains (2n, 2n+1, 2n+3) and (2n−1, 2n+1, 2n+3), whose largest vertex is 2n+3, outside the stated vertex set. Moreover, condition (1) for the jump 2n requires the edge (2n−2, 2n−1, 2n+1), which is not listed. Therefore I_n as defined is not a member of J_n, and the subsequent argument about forcing a blue I_n does not, as written, bear on the Ramsey number R(P_{n+2}, J_n).","section":"Section 2, upper bound, definition of I_n"},{"comment":"The proof's central step is the assertion “Using this fact we can check that β(u,v) > n would force a blue I_n.” No proof of this assertion is given, and it is the entire justification for the bound β(u,v) ≤ n, which is needed to define D(v) ⊂ [n]^2 and to obtain the 4^n bound. This is not a routine detail: it requires a construction mapping the witnessing sequence v_1 < ... < v_{2ℓ−1} to a copy of I_n and a verification that all required triples are blue using the inequality α(x,y) ≥ α(y,z). In addition, the definition of β is not self-contained: with a sequence v_1 < ... < v_{2ℓ−1}, the displayed condition references vertices v_{2i+2} and v_{2i+3} for values of i near ℓ that lie outside the sequence. Until the construction is supplied and the indexing is corrected, the upper bound R(P_{n+2}, J_n) ≤ 4^n · r(3;n) is unsupported.","section":"Section 2, upper bound, β definition and the missing check"}],"minor_comments":[{"comment":"The proof says the coloring contains no red P_n, but the theorem requires no red P_{n+2}. Since the α bound only needs the absence of P_{n+2}, this is likely a typo; it should be corrected to P_{n+2} throughout the upper-bound proof.","section":"Section 2, upper bound, first paragraph"},{"comment":"The range of the index i in the displayed condition for β should be made explicit (for example, i = 1, ..., ℓ−1 with all referenced indices lying within the sequence), so that the definition is unambiguous.","section":"Section 2, definition of β"},{"comment":"The expression “4n” should be “4^n” in the abstract, Theorem 1.3, and the upper-bound proof; the intended power is clear from the context but the missing superscript is a persistent typo.","section":"Throughout"},{"comment":"In the sentence “regard α as an n-coloring of pairs of S,” it would be clearer to specify the induced coloring of unordered pairs via (x,y) ↦ α(min{x,y}, max{x,y}).","section":"Section 2, upper bound, final argument"}],"recommendation":"major_revision","confidential_remarks":"The gaps in the proof are substantial. The lower bound appears repairable by adjusting the induction, but the upper bound requires a genuinely missing proof of the β-to-blue-I_n implication, and the definition of I_n needs to be corrected. If the authors cannot supply a complete verification of that implication, the main theorem will remain unsupported. I recommend major revision to give them the opportunity to fill these gaps, but the final decision should depend on whether the missing construction and proof are actually provided."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's real contribution is a new two-sided relation: r(3;n) ≤ R(P_{n+2}, J_n) ≤ 4^n · r(3;n), where J_n is a newly invented family of ordered 3-uniform hypergraphs ('monotone paths with n jumps'). If correct, this ties Erdős's old question on whether r(3;n) is exponential to a hypergraph Ramsey problem. That is a genuinely fresh angle, not a repackaging of known results. The lower bound is the cleaner half: the induction argument is coherent, the base case works, and the reduction from an n-coloring of pairs to a red/blue coloring of triples is clever. I believe that part is solid.\n\nThe upper bound, however, is not convincing as written. The key step is the sentence 'Using this fact we can check that β(u,v) > n would force a blue I_n inside K_N^{(3)}.' That check is not shown. It is the only step that gives β(u,v) ≤ n, which is what bounds D(v) ⊂ [n]^2 and produces the 4^n factor. Without it, the upper bound does not follow. On top of that, the definition of I_n is genuinely mis-specified: it is declared on [2n+1], but for i = n the listed triples (2n, 2n+1, 2n+3) and (2n-1, 2n+1, 2n+3) require vertices outside that set, and condition (1) for the last jump similarly needs 2n+2. These are not harmless typos; they make the asserted check impossible to verify as written. There is also a smaller mismatch: the upper-bound proof assumes 'no red P_n' while the theorem is about P_{n+2}; this is harmless (no red P_n is a stronger assumption), but it adds to the sense that the text was not carefully proofread.\n\nThe circularity concern the reader raised is not an issue: r(3;n) is an independent benchmark, and J_n is a new combinatorial object, not a fitted artifact. The authors are also appropriately candid about which parts are inspired by Mubayi–Suk and Moshkovitz–Shapira.\n\nBottom line: the lower bound alone is worth knowing, and the equivalence would be a nice result if the upper bound can be completed. Right now the main theorem is not fully established. The fix is probably local, but it is not present. This paper deserves a serious referee: a good referee can push the authors to supply the missing construction, or find a counterexample. I would send it to review, with a request that the upper bound be rewritten and the indexing corrected.\n\nWho is this for: researchers in Ramsey theory, especially those working on multicolor triangle Ramsey numbers or ordered hypergraphs. A reading group could compare this with the Mubayi–Suk and Moshkovitz–Shapira papers to test whether the gap is easily fillable.","headline":"New equivalence between r(3;n) and hypergraph Ramsey for a new family, but the upper bound rests on an unproved and mis-indexed claim.","tokens_in":6301,"tokens_out":2353,"would_cite":false,"duration_ms":22355,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D10","05C65","05C55","05A17"],"pacs":[],"model":"deepseek-v4-flash","headline":"This note proves that the multicolor triangle Ramsey number r(3;n) and the ordered-hypergraph Ramsey number R(P_{n+2},J_n) are within a 4^n factor of each other, making the exponential-growth question for r(3;n) equivalent to a…","keywords":["multicolor Ramsey numbers","triangle Ramsey numbers","ordered hypergraphs","monotone paths","3-uniform hypergraphs","Ramsey equivalence","integer partitions","jumps"],"falsifier":"For a small n such as n=2 or n=3, directly search over red/blue colorings of the complete ordered 3-uniform hypergraph on more than 4^n·r(3;n) vertices and check whether every coloring avoiding a red P_{n+2} still contains a blue copy of I_n; a counterexample would refute the upper bound. Also check the internal consistency of I_n by listing its edges for i=n, since the triple (2n, 2n+1, 2n+3) lies outside the declared vertex set [2n+1].","tokens_in":5144,"feed_emoji":"🔺","tokens_out":11442,"duration_ms":97887,"temperature":0.7,"pith_summary":"This note connects a century-old question in Ramsey theory to a new family of ordered 3-uniform hypergraphs. The main result is the two-sided inequality r(3;n) ≤ R(P_{n+2}, J_n) ≤ 4^n · r(3;n), where r(3;n) is the multicolor Ramsey number for triangles and R(P_{n+2}, J_n) is the Ramsey number of an ordered monotone path against the family of monotone paths with n jumps. Because the two quantities are within a fixed exponential factor of one another, deciding whether r(3;n) grows exponentially is exactly the same decision as whether the hypergraph parameter grows exponentially. A sympathetic reader can therefore replace the old pair-coloring problem with an ordered-hypergraph problem that may be easier to attack.","feed_headline":"Two Ramsey numbers proven equivalent up to a 4^n factor","feed_subtitle":"The paper's two-sided inequality lets the decades-old triangle problem be attacked via ordered 3-uniform hypergraphs.","key_machinery":"The proof uses two constructions. For the lower bound, given an n-coloring χ of pairs with no monochromatic triangle, color a triple (u,v,w) red if χ(u,v) < χ(v,w) and blue otherwise; a red P_{n+2} would require an increasing chain of n+1 colors, and the absence of a blue member of J_n is shown by an induction on the largest jump vertex that produces a monochromatic triangle in an auxiliary graph. For the upper bound, the proof fixes a specific hypergraph I_n ∈ J_n on vertices [2n+1] with edges consisting of all consecutive triples plus the three jump-pattern triples, and defines for every pair (u,v) an integer α(u,v) (the length of the longest red monotone path ending at the pair) and β(u,v) (the length of a certain alternating red/blue sequence). The pair (α(u,v), β(u,v)) lies in a downward-closed subset D(v) ⊂ [n]^2, and the number of such subsets is binomial(2n,n) ≤ 4^n, so pigeonhole produces r(3;n) vertices with the same D(v); a monochromatic triangle under α then extends a β-sequence and gives a contradiction.","core_discovery":"The central claim, Theorem 1.3, is that for every positive integer n, r(3;n) ≤ R(P_{n+2}, J_n) ≤ 4^n · r(3;n). Here P_{n+2} is the ordered 3-uniform monotone path on n+2 vertices whose edges are the consecutive triples, and J_n is the collection of ordered 3-uniform hypergraphs that contain all consecutive triples and admit a set of n jump vertices satisfying Conditions (0)–(2), namely no consecutive or endpoint jumps, two specified jump triples, and a closure condition on consecutive jumps. The left inequality comes from converting any n-coloring of pairs with no monochromatic triangle into a red/blue coloring of triples, and the right inequality comes from a counting argument on downward-closed sets associated to each vertex. The paper concludes that whether r(3;n) is exponential in Θ(n) is equivalent to whether R(P_{n+2}, J_n) is exponential in Θ(n).","pith_inferences":["The paper leaves implicit that the equivalence is not merely suggestive but mathematically two-sided; any method that bounds R(P_{n+2}, J_n) from above or below immediately transfers to r(3;n), up to the factor 4^n.","A natural next step, not taken here, is to enumerate downward-closed subsets more sharply or to exploit the special structure of J_n to reduce the 4^n factor; this would strengthen the correspondence without changing the proof architecture.","For small n such as n=2 or n=3, a direct computation of R(P_{n+2}, J_n) could test the asserted but unproved step that a large β-value forces a blue I_n, since the relevant hypergraphs are small enough to search.","The authors' concluding remark suggests an alternative route: instead of proving exponentiality of R(P_n, J_n), one might look inside n-colored complete graphs for families of non-increasing triples that become monotone paths after a suitable ordering, reframing the problem as a structural graph-theory question."],"forward_implications":["If R(P_{n+2}, J_n) is super-exponential in n, then so is r(3;n), settling the old question negatively; if R(P_{n+2}, J_n) is exponential, then r(3;n) is exponential, settling it positively.","By Corollary 1.4, r(3;n) ≤ R(P_{n+2}, P^4_{3n}), so the question reduces to the Ramsey number of a monotone path against a fourth-power path.","By Theorem 3.1, the same argument gives r(m;n) ≤ R(P_{n+2}, P^{m+1}_{mn}) for all m, n ≥ 3, extending the equivalence from triangles to larger cliques.","The proof shows that to prove r(3;n) is super-exponential, it suffices to prove R(P_n, I_n) is super-exponential for the single explicit hypergraph I_n defined in the upper-bound proof.","Any improvement in the counting of downward-closed subsets would directly improve the factor 4^n in the upper bound, since that factor comes entirely from that counting step."],"supporting_citations":[{"why":"It supplies the lower-bound technique of coloring triples by comparing pair colors, which turns a blue member of J_n into a monochromatic triangle in an auxiliary graph.","marker":"[10]"},{"why":"It provides the bijection between downward-closed subsets of [n]^2 and integer partitions, used to bound their number by binomial(2n,n) ≤ 4^n in the upper bound.","marker":"[9]"},{"why":"It supplies the monotonicity results that underpin the counting argument for downward-closed sets in the upper bound.","marker":"[4]"}],"fun_headline_variants":["Triangle Ramsey exponential iff path Ramsey exponential","Two Ramsey numbers equivalent up to factor 4^n","Exponential triangle conjecture linked to monotone paths","Ramsey links: triangles and 3-uniform paths","4^n equivalence: triangle and path Ramsey numbers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The upper-bound argument stands on the asserted but unproved step that if the auxiliary parameter β(u,v) exceeds n then a blue copy of I_n is forced; because the definition of I_n as written also appears to involve vertices outside its declared vertex set, that step is not yet established.","fun_headline_variants_meta":{"raw":{"variants":["Triangle Ramsey exponential iff path Ramsey exponential","Two Ramsey numbers equivalent up to factor 4^n","Exponential triangle conjecture linked to monotone paths","Ramsey links: triangles and 3-uniform paths","4^n equivalence: triangle and path Ramsey numbers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000533,"raw_usage":{"total_tokens":2560,"prompt_tokens":940,"completion_tokens":1620,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":1549}},"tokens_in":556,"tokens_out":1620,"duration_ms":11846,"temperature":1.0,"reasoning_tokens":1549,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:08:13.422771+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a small n such as n=2 or n=3, directly search over red/blue colorings of the complete ordered 3-uniform hypergraph on more than 4^n·r(3;n) vertices and check whether every coloring avoiding a red P_{n+2} still contains a blue copy of I_n; a counterexample would refute the upper bound. Also check the internal consistency of I_n by listing its edges for i=n, since the triple (2n, 2n+1, 2n+3) lies outside the declared vertex set [2n+1].","supporting_citations":[{"cited_title":"Mubayi and A","cited_arxiv_id":null,"evidence_quote":"It supplies the lower-bound technique of coloring triples by comparing pair colors, which turns a blue member of J_n into a monochromatic triangle in an auxiliary graph."},{"cited_title":"Moshkovitz and A","cited_arxiv_id":null,"evidence_quote":"It provides the bijection between downward-closed subsets of [n]^2 and integer partitions, used to bound their number by binomial(2n,n) ≤ 4^n in the upper bound."},{"cited_title":"Chv´ atal and J","cited_arxiv_id":null,"evidence_quote":"It supplies the monotonicity results that underpin the counting argument for downward-closed sets in the upper bound."}],"review_version":1}