{"id":"30a3f1ad-371b-4a71-83e9-fac8ecf81c70","arxiv_id":"2411.13416","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proposes bounds on the induced triangle Ramsey number R^Δ_ind(F), but the proofs of the two main bounds rely on a false Ramsey inequality and an insufficient extremal count.","lead":"This paper defines a Ramsey number for induced subgraphs whose triangles are monochromatic and claims new upper bounds for three families of graphs. The two main proofs contain apparent mathematical errors, so the headline results are not established as written.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 rests on the false bound R_2(n−1)+1 < 4n in Lemma 3.3; the required clique size is exponential in n, so the proof's iteration cannot run.","rationale":"The reader's weakest-assumption analysis is correct: the load-bearing failure is the false numerical bound on ℓ in Lemma 3.3. The bound is not an inessential optimization; the entire Erdős–Rado iteration in that lemma depends on r staying below sqrt((1/2) log m). With the true value of ℓ being exponential in n, the iteration cannot be executed. I find no way to repair this within the stated parameters, because the final step K_{\\ell−1} → (K_{n−1})^2 forces ℓ−1 ≥ R_2(n−1), so any valid ℓ is already exponential. I also checked the claimed bound in Theorem 1.2 and the count of blue copies there appears too small by a large factor to trigger Theorem 5.1, so that proof has a second, independent gap. Proposition 1.3, by contrast, is a self-contained polynomial argument and seems coherent. Thus I agree with the REJECT verdict: the central quantitative claims of the paper are not established, although the paper is not without merit. The concern is purely mathematical and would be settled by the explicit computation described above.","tokens_in":18299,"tokens_out":11539,"duration_ms":116533,"concrete_test":"One decisive check is to evaluate the asserted inequality in Lemma 3.3 at n=14. Using the classical Erdős lower bound R_2(13) ≥ 2^{13/2} > 90, we get ℓ = R_2(13)+1 ≥ 92. The paper's bound would require ℓ < 4n = 56 and ℓ ≤ sqrt((1/2) log_2 m) = sqrt(168) ≈ 12.96, so the premise fails. To see the downstream effect, run the greedy construction from the proof of Lemma 3.3 with r=92: after r steps the available set has size at most 2^{-92^2} · 2^{336}, far below sqrt(m)=2^{168}, so the density bound (3.3) no longer gives a vertex with degree ≥ |S_r|/4 and the induction cannot continue. If this computation is confirmed, Corollary 3.4, Lemma 3.6(A), and Theorem 1.1 do not follow from the stated proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 3.3 (Section 3.2), the proof sets ℓ = R_2(n−1)+1 and asserts ℓ < 4n and hence ℓ ≤ sqrt((1/2) log m) for m = 2^{24n}. The symbol R_2(n−1) is the diagonal graph Ramsey number, for which R_2(k) ≥ 2^{k/2}. Consequently ℓ grows exponentially in n, while 4n and sqrt((1/2) log m) are subexponential. The claim is already false at n = 14: R_2(13) ≥ 91, so ℓ ≥ 92, but 4n = 56 and sqrt((1/2) log_2(2^{336})) = sqrt(168) ≈ 12.96. This inequality is not a minor technicality: it is exactly the condition that keeps the current set S_r at size at least sqrt(m) throughout the greedy construction of the clique x_1,...,x_ℓ. Once r would need to reach the true value of ℓ, the lower bound |S_r| ≥ (1/2)^{r^2} m falls below m^{1/2}, so the density property (3.3) can no longer be invoked. Hence the Erdős–Rado-style argument cannot produce even the auxiliary clique needed for the final application of K_{\\ell−1} → (K_{n−1})^2. Corollary 3.4 and Lemma 3.6(A) depend directly on Lemma 3.3, so Theorem 1.1 is unsupported. The problem is internal to the proof, not a matter of disagreement with known bounds.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines the induced triangle Ramsey number R_ind^Δ(F), the smallest N such that some graph G on N vertices has the property that every 2-coloring of the triangles of G yields an induced copy of F whose triangles are all monochromatic. The main results are three quantitative upper bounds: a tower-type bound R_ind^Δ(F) ≤ 2^{2^{240 n}} when the triangle hypergraph K_3(F) is linear (Theorem 1.1), a double-exponential bound R_ind^Δ(F) ≤ 2^{2^{c n log n}} for graphs in the family B_n (Theorem 1.2), and a polynomial bound R_ind^Δ(F) ≤ n^4 when K_3(F) is a tight tree (Proposition 1.3). The proofs combine random graph properties, a weak hypergraph regularity lemma, and an embedding lemma for linear 3-graphs.","tokens_in":18572,"tokens_out":18086,"duration_ms":169173,"significance":"If the theorems are correct, they would provide the first explicit quantitative bounds for R_ind^Δ(F) for these natural families, including a polynomial bound in the tight-tree case. The paper builds a useful framework: Lemma 3.6 shows that a large random graph contains many monochromatic cliques with prescribed regularity, Lemma 4.2 gives a weak regularity lemma for n-partite n-uniform hypergraphs, and Lemma 4.3 is an embedding lemma for linear 3-graphs. Proposition 1.3's approach via average triangle degree is elegant and appears salvageable. However, the two main theorems currently rest on incorrect numerical estimates, so the central claims are not established as written.","major_comments":[{"comment":"The proof of Lemma 3.3 relies on the assertions ℓ = R_2(n−1)+1 < 4n and ℓ ≤ sqrt((1/2) log m). Since R_2(k) is the diagonal graph Ramsey number and R_2(k) ≥ 2^{k/2}, the inequality ℓ < 4n is false for all sufficiently large n; for instance, at n = 14, R_2(13) ≥ 91 while 4n = 56. More importantly, with the stated m = 2^{24n} (or any single-exponential m), sqrt((1/2) log m) is polynomial in n, whereas the true ℓ is exponential. The inductive construction requires |S_r| ≥ m^{1/2} to invoke property (3.3) at every step r < ℓ; once r reaches the true exponential value of ℓ, the bound |S_r| ≥ (1/2)^{r^2} m falls far below m^{1/2}. Consequently the auxiliary clique of size ℓ cannot be produced, and Corollary 3.4 and Lemma 3.6(A) are unsupported. To repair the proof, the authors need to replace this estimate with a correct upper bound on R_2(n−1) and choose m (and N) so that sqrt((1/2) log m) ≥ R_2(n−1)+1; the current numerical constants do not do this.","section":"§3.2, Lemma 3.3"},{"comment":"The counting argument for the number of distinct monochromatic copies of F[A] in the blow-up is incorrect. From |T| = N^m/(2 binom(m,k)) and the fact that each F[A]-copy is contained in at most N^{m-k} transversals, the number of distinct copies is at least N^k/(2 binom(m,k)) ≈ k! N^k/(2 m^k). With the stated choice N = m k^{2k}, this lower bound is ≈ k! k^{2k^2}/2, a constant depending only on k and n, not on N. Theorem 5.1 with t = 2 requires the number of copies to exceed C N^{k − 1/2^{k−1}}, which is enormous for the chosen N; the count is too small by a factor roughly N^{1/2^{k−1}} times a constant. The displayed lower bound in the manuscript does not follow from the preceding estimate. The construction needs a substantially larger N (e.g., N ≥ C m^{k 2^{k−1}} for a suitable constant C) for the extremal step to work, and this must be reconciled with the claimed final bound of 2^{2^{c n log n}}.","section":"§5, proof of Theorem 1.2"}],"minor_comments":[{"comment":"The notation \"m = 224n\" is ambiguous; it should be clarified whether m denotes 2^{24n} or 2^{2^{4n}}. The subsequent inequalities depend on this distinction.","section":"§3.2, Lemma 3.3"},{"comment":"The displayed lower bound \"≥ 1/2 (N/m)^k N^{k−1/2^{k−1}}\" is not the expression obtained from the preceding line; it appears to be a typesetting or algebraic error.","section":"§5, proof of Theorem 1.2"},{"comment":"Event (B) is stated for sets S with |S| ≥ n, but the proof applies it to sets with |S| ≤ n; the statement should read |S| ≤ n for the argument to be coherent.","section":"§6, proof of Proposition 1.3"},{"comment":"Condition (2) omits its subject: it should say that the triples (V_i, V_j, V_k) are (ε,d)-dense in H for every 1 ≤ i < j < k ≤ n.","section":"§4.2, Lemma 4.3"},{"comment":"The manuscript contains several OCR-like artifacts that obscure the mathematics, for example \"ei /integerdividetviu\", \"rmspkq\", and \"pε,d q-regular\". A careful proofreading pass is needed before publication.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The two main theorems are not proved as written, but the errors appear repairable with adjusted parameters: Lemma 3.3 requires replacing the false bound ℓ < 4n with a correct Ramsey-number estimate and an appropriate choice of m, and Theorem 1.2 requires a substantially larger N for the extremal counting step. Because the proof strategies and auxiliary lemmas are largely sound, I recommend major revision rather than rejection. The authors must, however, correct these load-bearing numerical issues before the paper can be considered for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper defines a natural Ramsey number R^Δ_ind(F) and gives three quantitative bounds. The definition itself is reasonable, and Proposition 1.3 (polynomial bound for tight trees) looks correct. But the two main theorems, 1.1 and 1.2, rest on concrete false inequalities, so the paper is not publishable in its current form.\n\nThe first problem is in Lemma 3.3, where the proof sets ℓ = R_2(n−1)+1 and asserts ℓ < 4n. That is false: the diagonal graph Ramsey number is exponential in n, so ℓ is far larger than 4n for n ≥ 14. The later bound ℓ ≤ sqrt((1/2) log m) is used to keep the shrinking set S_r above m^{1/2}; with the true ℓ the iteration cannot run. Theorem 1.1 depends on this lemma, so it is unsupported.\n\nThe second problem is in Theorem 1.2. The proof counts blue copies of F[A] and gets about (1/2)(N/m)^k, then tries to apply Erdős's extremal bound for K^{(k)}_{2,...,2}, which requires more than C N^{k−1/2^{k−1}} copies. Since N = m k 2^k, the count is smaller by roughly N^{1/2^{k−1}} than needed. The double-exponential upper bound therefore does not follow from the argument.\n\nThere is good material here: the weak-regularity framework and the embedding lemma in Appendix C are sensible, and the paper is clearly written. Proposition 1.3 appears sound, and the definition of R^Δ_ind is a natural object worth studying. But the two main theorems are the core of the paper, and their proofs contain load-bearing errors that are not minor typos.\n\nI would not send this to peer review in its current state. A referee would quickly find the same failures. The authors need to rethink the proofs, especially the clique-construction in Lemma 3.3 and the counting in Theorem 1.2. If those are repaired, the paper could be a solid contribution. For now, it is not.","headline":"Two of the three main theorems rest on false inequalities, so the paper's headline bounds are unsupported; the definition and Proposition 1.3 are fine, but the main proofs need serious rework.","tokens_in":19168,"tokens_out":4405,"would_cite":false,"duration_ms":40999,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C55","05C65","05D10","05C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes the first quantitative upper bounds for the induced triangle Ramsey number $R_{\\mathrm{ind}}^{\\Delta}(F)$, including a tower-type bound for graphs whose triangles form a linear hypergraph.","keywords":["induced Ramsey numbers","triangle colorings","triangle hypergraph","linear hypergraphs","tight trees","weak hypergraph regularity","random graphs","monochromatic cliques"],"falsifier":"In the proof of Lemma 3.3, compare the claimed bound $\\ell = R_2(n-1)+1$ with the requirement $\\ell \\le \\sqrt{(1/2)\\log m}$ for $m = 2^{24n}$; since $R_2(n-1)$ grows exponentially while the threshold is $O(\\sqrt{n})$, the chain of inequalities fails for $n \\ge 2$, and the covering argument that selects $\\ell$ vertices cannot run at that size. This calculation settles that the proof of Theorem 1.1, as written, does not establish the tower-type bound.","tokens_in":18034,"feed_emoji":"🔺","tokens_out":10722,"duration_ms":103563,"temperature":0.7,"pith_summary":"For every graph $F$ there is a host graph $G$ such that any red/blue coloring of the triangles of $G$ contains an induced copy of $F$ whose triangles are all one color. This paper defines $R_{\\mathrm{ind}}^{\\Delta}(F)$, the smallest number of vertices of such a $G$, and proves the first numerical upper bounds for it in three natural settings. If $K_3(F)$, the 3-uniform hypergraph of triangles of $F$, is linear, the paper claims $R_{\\mathrm{ind}}^{\\Delta}(F) \\le 2^{2^{240n}}$. If $F$ belongs to the family $B_n$---one part triangle-free, the other part independent, arbitrary cross edges---the bound is double exponential, $2^{2^{c n \\log n}}$. If $K_3(F)$ is a tight tree, the bound is polynomial, $n^4$. These are the first concrete bounds for this Ramsey function, which previously was known only to be finite.","feed_headline":"Double-exponential host graph forces monochromatic triangles","feed_subtitle":"For triangle-linear graphs, a host of double-exponential size forces the desired copy; tight trees need only n^4.","key_machinery":"The central object is the 3-uniform triangle hypergraph $K_3(F)$, whose edges are the vertex triples that form triangles in $F$; a suitable copy must be strongly induced, meaning that no hyperedge outside the copy contains any pair from the copy. The proofs build host graphs mostly from the random graph $G(N,1/2)$: a density property forces near-uniform edge densities on all large pairs; a greedy covering argument produces many monochromatic $K_n$ cliques; weak hypergraph regularization produces $(\\varepsilon,d)$-dense tuples; and an embedding lemma for linear 3-graphs places a copy of $F$. For $B_n$, the construction is a blow-up of an induced Ramsey graph with added universal vertices, and for tight trees a minimal-subgraph argument keeps red-triangle degree high to permit greedy embedding.","core_discovery":"The paper's central discovery is that the existence theorem $G \\indarrow (F)^{\\Delta}$ can be made quantitative for three families of $F$, and that the natural obstruction is the structure of the triangle hypergraph $K_3(F)$. The main theorem says that whenever $K_3(F)$ is linear---no two triangles share an edge---a host graph on $2^{2^{240n}}$ vertices suffices. For $F$ in $B_n$ the paper builds an explicit host graph from a blow-up of an induced Ramsey graph plus a layer of universal vertices, giving $2^{2^{c n \\log n}}$. For tight trees it gives $n^4$. The proofs combine random graphs, weak hypergraph regularity, and embedding lemmas, and the paper notes the same strategy extends to colorings of $r$-cliques for $r \\ge 3$.","pith_inferences":["The double-exponential bound for $B_n$ and the polynomial bound for tight trees do not use Lemma 3.3, so they remain intact even if the tower-type proof needs repair.","A repair of Theorem 1.1 would likely replace the greedy clique-selection step by a different concentration argument or increase the parameter $m$, leaving the weak-regularity embedding pipeline unchanged.","The same three-step scheme suggests that the function $R_{\\mathrm{ind}}^{K_r}(F)$ should admit bounds of tower height $r-1$, and that the tight-tree result may extend to any graph whose triangle hypergraph has bounded tree-depth."],"forward_implications":["For every $n$-vertex graph $F$ with linear $K_3(F)$, there exists a host graph on $2^{2^{240n}}$ vertices such that every 2-coloring of its triangles contains a monochromatic induced copy of $F$.","For every $F$ in $B_n$, the same statement holds with $2^{2^{c n \\log n}}$ vertices.","For every $F$ whose $K_3(F)$ is a tight tree, $n^4$ vertices suffice.","Because $R_{\\mathrm{ind}}(F) \\le R_{\\mathrm{ind}}^{\\Delta}(F)$, all three bounds transfer to ordinary induced Ramsey numbers for these families."],"supporting_citations":[{"why":"supplies the greedy covering argument that Lemma 3.3 uses to build a clique whose triangle colors depend only on the first pair","marker":"[8]"},{"why":"the weak hypergraph regularity and linear-hypergraph embedding lemma that Lemma 4.3 extends","marker":"[14]"},{"why":"the density-increment weak regularization argument that Lemma 4.2 extends to n-partite n-uniform hypergraphs","marker":"[20]"},{"why":"the extremal bound for complete k-partite k-graphs used in the proof of Theorem 1.2 to force many blue copies","marker":"[9]"},{"why":"the polynomial-size induced Ramsey construction for trees whose greedy embedding idea Section 6 adapts to tight trees","marker":"[2]"},{"why":"the exponential induced Ramsey upper bound used in Theorem 1.2 to select the graph H in the blow-up construction","marker":"[5]"}],"fun_headline_variants":["Double-exponential host forces monochromatic triangles","Tight trees need only n^4 for induced triangle Ramsey","Induced triangle Ramsey: from double-exponential to n^4","Coloring triangles to force any graph: new host bounds","Triangle colorings force induced copies: host sizes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The tower-type bound depends on the assertion that the graph-Ramsey number $R_2(n-1)$ is less than $4n$, which is false for $n \\ge 2$, so the step that forces many monochromatic $K_n$ cliques in the random host graph cannot be carried out at the stated size $m = 2^{24n}$.","fun_headline_variants_meta":{"raw":{"variants":["Double-exponential host forces monochromatic triangles","Tight trees need only n^4 for induced triangle Ramsey","Induced triangle Ramsey: from double-exponential to n^4","Coloring triangles to force any graph: new host bounds","Triangle colorings force induced copies: host sizes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000919,"raw_usage":{"total_tokens":3875,"prompt_tokens":808,"completion_tokens":3067,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":424,"completion_tokens_details":{"reasoning_tokens":2988}},"tokens_in":424,"tokens_out":3067,"duration_ms":21388,"temperature":1.0,"reasoning_tokens":2988,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:28:18.998400+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the proof of Lemma 3.3, compare the claimed bound $\\ell = R_2(n-1)+1$ with the requirement $\\ell \\le \\sqrt{(1/2)\\log m}$ for $m = 2^{24n}$; since $R_2(n-1)$ grows exponentially while the threshold is $O(\\sqrt{n})$, the chain of inequalities fails for $n \\ge 2$, and the covering argument that selects $\\ell$ vertices cannot run at that size. This calculation settles that the proof of Theorem 1.1, as written, does not establish the tower-type bound.","supporting_citations":[{"cited_title":"Erd˝ os and R","cited_arxiv_id":null,"evidence_quote":"supplies the greedy covering argument that Lemma 3.3 uses to build a clique whose triangle colors depend only on the first pair"},{"cited_title":"Kohayakawa, B","cited_arxiv_id":null,"evidence_quote":"the weak hypergraph regularity and linear-hypergraph embedding lemma that Lemma 4.3 extends"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the density-increment weak regularization argument that Lemma 4.2 extends to n-partite n-uniform hypergraphs"},{"cited_title":"Erdős, On extremal problems of graphs and generalized graphs , Israel J","cited_arxiv_id":null,"evidence_quote":"the extremal bound for complete k-partite k-graphs used in the proof of Theorem 1.2 to force many blue copies"},{"cited_title":"Beck, On size Ramsey number of paths, trees and circuits","cited_arxiv_id":null,"evidence_quote":"the polynomial-size induced Ramsey construction for trees whose greedy embedding idea Section 6 adapts to tight trees"},{"cited_title":"Conlon, J","cited_arxiv_id":null,"evidence_quote":"the exponential induced Ramsey upper bound used in Theorem 1.2 to select the graph H in the blow-up construction"}],"review_version":1}