{"id":"a933e79c-d2b7-490b-bf83-582046cbc49a","arxiv_id":"2507.13329","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"The minimum number of colors in a (C_{2k},3)-coloring of K_{n,n} is exactly (7/20)n+o(n) for k=3 and lies between improved explicit bounds for all k≥4.","lead":"The paper finds the exact asymptotic number of colors needed to color the edges of a complete bipartite graph so that no six-cycle is made of only two colors, and it improves the known bounds for longer even cycles. This answers an open question of Lane and Morrison and gives a sharper benchmark for generalized Ramsey numbers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 is stated for p+q=k, but the paper's H1 edges contain (k-1)(2k-1) P-vertices and O(k^2) Q-vertices; without explicitly renaming the theorem's parameter, the black-box matching theorem does not clearly cover the construction.","rationale":"The reader's weakest-assumption analysis identifies the same point: the upper bound is a black-box application of Theorem 3.1, and the parameter relation p+q=k is not reconciled with H1 edges that contain O(k^2) vertices from P union Q. This concern is load-bearing because the entire upper-bound construction depends on the existence of a P-perfect matching avoiding C and D. The lower-bound argument in Section 2 appears internally sound: the stripping counts are correct, the pair-disjointness inequality (8) follows from the two-vertex sharing argument, and the k=3 specialization to h(1/2)=7/10 is valid. The conflict-system checks in Sections 3.3-3.5 are asymptotic and mostly plausible, but they are all conditional on being inside the theorem's hypotheses. If p+q is merely a dummy parameter and the theorem extends to p+q=4k^2-3k, the proof can be repaired by a short restatement; if not, the central upper bound fails. Therefore the appropriate verdict is conditional acceptance: the authors must explicitly verify and state that Theorem 3.1 applies with the actual edge size of their H1, or define a renamed parameter K and adjust the constant choices accordingly.","tokens_in":15001,"tokens_out":55480,"duration_ms":612886,"concrete_test":"Consult the exact statement and proof of Theorem 3.1 in Joos-Mubayi-Smith [16] and determine whether it applies to arbitrary fixed p+q (e.g., K = 4k^2 - 3k) or only to p+q equal to the cycle parameter k. If it applies to arbitrary fixed p+q, restate the application in Section 3 with K in place of the theorem's k, choose epsilon < min(1/K, 2/(2K-delta)), and verify all conditions (H1)-(H4), (C1)-(C3), (D1)-(D4) still hold with d as in (12). If it does not apply to arbitrary p+q, the upper bound in Theorem 1.1 is unsupported because the construction's H1 edges have size O(k^2), not k.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Section 3.2.1, an H1 edge e_{K,c} is defined with P = E(K_{n,n}) and Q = {v_c : v in V, c in W} union EA union EB. For a biclique K_{k-1,2k-1}, the edge contains all |E(K)| = (k-1)(2k-1) graph edges as P-vertices, plus all 3k-2 color-vertices v_c and all C(2k-1,2) pairs from the large side, so p = 2k^2 - 3k + 1 and q = 2k^2 - 1. Hence p + q = 4k^2 - 3k, not the k appearing in Theorem 3.1's hypothesis 'For p+q = k >= 2'. The paper never states that the theorem's p+q is a free dummy parameter independent of the C_{2k} parameter, and the constant choice (11), 0 < delta << epsilon << 1/k, is not tied to the p+q value. If the theorem genuinely requires p+q to equal the cycle parameter, the P-perfect matching is not guaranteed and the upper-bound construction collapses. The initial size inequalities d^epsilon <= |P| = n^2 and |P union Q| = O(n^2) <= exp(d^{epsilon^3}) are also not spelled out; they are likely satisfiable for suitable epsilon, but the p+q mismatch is a substantive hypothesis-check rather than a cosmetic one.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the generalized Ramsey number r(K_{n,n}, C_{2k}, 3). It proves by a stripping argument a new lower bound for every k >= 4 and a sharp asymptotic value 7/20 n + o(n) for k = 3. For the upper bound it constructs a coloring via conflict-free hypergraph matchings, using as color classes appropriately packed bicliques K_{k-1,2k-1} with small sides in sparse linear hypergraphs S_A/S_B and large sides whose pairs avoid S_A/S_B; leftover edges are assigned fresh colors in a second stage. The conflict systems C and D are designed so that a C-union-D-free P-perfect matching corresponds to a coloring with no 2-colored C_{2k}.","tokens_in":15261,"tokens_out":42731,"duration_ms":473686,"significance":"If the proof is completed, the results are significant: they improve both known bounds for every k, give the first sharp asymptotic value for C_6, and answer a question of Lane and Morrison. The lower-bound argument is a clean self-contained weighting/stripping argument with explicit inequalities, and the upper-bound application of conflict-free matching is natural and technically detailed. The main issues are missing definitions and hypothesis checks rather than obvious mathematical errors.","major_comments":[{"comment":"The set W of first-stage colors is never given a cardinality. The total number of colors in the construction is |W| + |W'|, so the upper bound in Theorem 1.1 can only be verified from the stated computations if |W| = (3k-2)/(2(k-1)(2k-1)) n (or an equivalent quantity of order n). The display for d_{H1}(e) currently reads as if |W| were part of the constant 3k-2 divided by (2k-1)!; this makes the color count unverifiable and, if |W| were constant, the construction would contradict the lower bound. Please define W explicitly, correct the display, and confirm that d has the form (12). This is load-bearing because the color count is the statement being proved.","section":"Section 3.2.1 and Section 3.3"},{"comment":"The hypotheses of Theorem 3.1 are not explicitly checked in the setup. The constructed H1 has p=(k-1)(2k-1) P-vertices and q=(3k-2)+C(2k-1,2) Q-vertices per edge, so p+q=4k^2-3k, not the cycle parameter k appearing in the theorem's statement 'For p+q=k'. Since the theorem is being applied with a different value of its parameter (call it K=p+q), this renaming must be stated, and the size inequalities d^epsilon <= |P| <= |P union Q| <= exp(d^{epsilon^3}) should be verified for |P|=n^2, |P union Q|=Theta(n^2), and d=Theta(n^{2k-delta}) with epsilon sufficiently small. These are routine but currently omitted.","section":"Section 3.1 and Section 3.2"}],"minor_comments":[{"comment":"The assertion that if every c'-colored biclique shares two vertices with the single c-colored biclique then their small sides all coincide is not immediate; it uses linearity of S_A and S_B together with the condition (Z choose 2) subseteq E_A union E_B. Please add the one-sentence justification, since this is the step that prevents a two-biclique 2-colored cycle.","section":"Section 3.2.2, Claim 3.4"},{"comment":"The parameter ell in Theorem 3.1 is not assigned in the application; one should take ell = 2k to match the bounds in (C1) and (D1).","section":"Section 3.1"},{"comment":"In the text following the k=3 lower bound, the notation f(K_{n,n}, C_6, 3) is used instead of r(K_{n,n}, C_6, 3); please make the notation consistent.","section":"Section 2"}],"recommendation":"major_revision","confidential_remarks":"I found no fatal flaw in the main construction; the apparent counterexample with two opposite-orientation bicliques is excluded by the pair condition (Z choose 2) subseteq E_A union E_B. The revision should focus on making the application of Theorem 3.1 fully explicit, especially the definition of W and the identification of p and q."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a good paper. It answers Lane and Morrison's question, proves the first asymptotically sharp constant for C6, and improves both bounds for all k≥4. The k=3 lower bound is self-contained and clean; the upper bound is a careful application of the JMS conflict-free matching theorem with a genuine new idea: forcing the packed bicliques to share their (k−1)-sides, which is what buys the better constant. The paper deserves serious refereeing.\n\nWhat is actually new: the packing trick is real and accounts for the improvement; the paper also removes the floor in the denominator that Lane and Morrison flagged. The lower bound argument, while elementary, is done with care, and the k=3 specialization via monotonicity of h(a) is a neat touch.\n\nSoft spots: the stress-test concern about Theorem 3.1's p+q=k is not a real problem. The theorem's k is a dummy parameter: you instantiate it with the actual edge size in H1, which is Θ(k^2) here. The paper should have renamed it to avoid confusion, and the ε in (11) technically needs to be chosen with respect to that larger parameter, but that is a one-line fix. The initial size inequalities (d^ε ≤ |P|, |P∪Q| ≤ exp(d^{ε^3})) are clearly satisfiable for the stated d, though the paper does not spell them out. So the concern is expository, not structural. The dense asymptotic counts in Sections 3.3–3.5 are standard for this method; I found no circularity or fitted constants. The reliance on the external theorem from [16] is explicit and appropriate.\n\nWho this is for: anyone working in generalized Ramsey theory or conflict-free hypergraph matchings. It is a solid application that extends the method, and the conjecture at the end is plausible. I would cite it.\n\nRecommendation: send it to a good combinatorics journal; a serious referee will want the parameter renaming fixed and a bit more detail on the auxiliary hypotheses, but the main result is correct as far as I can see.","headline":"Solid paper: settles the C6 asymptotic and improves bounds for all longer cycles; the p+q=k worry in the stress test is a red herring, not a real gap.","tokens_in":15813,"tokens_out":2941,"would_cite":true,"duration_ms":30436,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C15","05C55","05C65","05D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that coloring the edges of $K_{n,n}$ so that no 6-cycle uses only two colors requires asymptotically $\\frac{7}{20} n$ colors, and improves the bounds for all longer even cycles.","keywords":["generalized Ramsey number","bipartite Ramsey theory","even cycles","conflict-free hypergraph matchings","edge-coloring","random sparse hypergraphs","K_{n,n}"],"falsifier":"Check the parameter relation $p+q=k$ in Theorem 3.1 against the actual number of $P$ and $Q$ vertices in a generic $\\mathcal{H}_1$ edge; if the relation cannot be met, the upper-bound proof does not go through. A complementary check is to verify the inequalities (H1)--(H4), (C1)--(C3), and (D1)--(D4) numerically for a fixed small $k$, such as $k=4$, using $d$ as defined in equation (12).","tokens_in":14741,"feed_emoji":"🎨","tokens_out":8068,"duration_ms":86010,"temperature":0.7,"pith_summary":"The paper studies the generalized Ramsey number $r(K_{n,n}, C_{2k}, 3)$, the minimum number of colors needed to color the edges of the complete bipartite graph $K_{n,n}$ so that every copy of the even cycle $C_{2k}$ uses at least three colors. It establishes that for $k=3$ this number is asymptotically $\\frac{7}{20} n$, and for every $k \\ge 4$ it lies between an explicit lower bound and the upper bound $\\frac{3k-2}{2(k-1)(2k-1)} n + o(n)$. The upper bound answers a question of Lane and Morrison about removing an integer-floor obstruction from the previously known bound. The interest is that the lower and upper arguments are different in character, yet for six-cycles they meet exactly in the limit.","feed_headline":"Six-cycle Ramsey bound pinned at 0.35n colors","feed_subtitle":"Matching upper and lower bounds settle the six-cycle case and tighten all longer even cycles.","key_machinery":"The upper-bound construction is carried by the conflict-free hypergraph matching theorem of Joos, Mubayi and Smith (Theorem 3.1), which produces a matching in a hypergraph $\\mathcal{H} = \\mathcal{H}_1 \\cup \\mathcal{H}_2$ that covers all of $P$ and avoids forbidden submatchings called conflicts. The paper builds $\\mathcal{H}_1$ whose edges are colored bicliques $K_{k-1,2k-1}$ with the $(k-1)$-vertex side belonging to a random sparse linear hypergraph $S_A$ or $S_B$, obtained by randomly retaining edges of an $(n,k-1,2)$-Steiner system; the $b$-side of each biclique is chosen so that its vertex pairs avoid the edges of $S_A \\cup S_B$. A second hypergraph $\\mathcal{H}_2$ consists of spare colored edges that handle leftover pairs. The conflict systems $\\mathcal{C}$ and $\\mathcal{D}$ encode all ways a two-colored $C_{2k}$ could form, and the forced overlap of the small sides is the innovation that makes the packing more efficient.","core_discovery":"The paper's central claim is Theorem 1.1: for every $k \\ge 4$, $$\\frac{4\\sqrt{$k^{4}$-$6k^{3}$+$12k^{2}$-9k+2}-$4k^{2}$+12k-5}{2(k-2)} n \\le r(K_{n,n}, C_{2k}, 3) \\le \\frac{3k-2}{2(k-1)(2k-1)} n + o(n),$$ and for $k=3$, $$r(K_{n,n}, C_6, 3) = \\frac{7}{20} n + o(n).$$ The $k=3$ statement is an asymptotically sharp estimate, and the upper bound improves the earlier bound by a constant factor for large $k$, resolving the Lane--Morrison question. The insight behind the improvement is that colored bicliques $K_{k-1,2k-1}$ can be packed much more efficiently if their small sides are forced to lie inside a sparse linear hypergraph; two such bicliques may overlap without creating a two-colored $C_{2k}$, so fewer colors are needed.","pith_inferences":["Beyond the paper, the forced-overlap packing idea looks transferable: for any forbidden bipartite subgraph built from overlapping bicliques, confining the small sides of the bicliques to a sparse linear hypergraph could yield improved upper bounds.","The clean match at $k=3$ makes $k=4$ the natural next test of the conjecture; the new interval $[0.227n, 0.239n]$ is narrow enough that a sharper count on either side might close the gap.","A reader wanting to fully trust the upper bound should first reconcile the parameter relation $p+q=k$ in Theorem 3.1 with the $O(k^2)$ vertices from $P \\cup Q$ appearing in each $\\mathcal{H}_1$ edge; the paper does not spell this out, so this is an open technical point rather than an established failure."],"forward_implications":["For $k=3$, any valid coloring needs at least $\\frac{7}{20} n - o(n)$ colors and one exists with at most $\\frac{7}{20} n + o(n)$ colors, so the asymptotic answer for six-cycles is now known exactly.","For every $k \\ge 4$, the new bounds improve both the previous lower and upper bounds; for large $k$ the new upper bound is roughly $\\frac{3}{4}$ of the old one.","The paper conjectures that the upper bound is tight for all $k$, and proves this conjecture under the additional assumption that every monochromatic component is essentially a biclique $K_{k-1,b}$ with $b \\ge k$.","The upper-bound construction provides an explicit coloring scheme: most edges are covered by colored bicliques whose small sides lie in the sparse hypergraphs, and a tiny leftover set of edges is colored with extra colors."],"supporting_citations":[{"why":"Supplies the conflict-free hypergraph matching theorem (Theorem 3.1) that is the engine of the upper-bound construction.","marker":"[16]"},{"why":"Introduced the conflict-free matching approach for generalized Ramsey upper bounds, including the asymptotic $r(K_{n,n}, C_4, 3)$ result that the paper extends.","marker":"[15]"},{"why":"Gives the previous upper bound with the integer floor and poses the question about removing it, which this paper answers.","marker":"[17]"},{"why":"The authors' earlier joint paper with Heath and Zerbib gives the previous lower bound in (1) and related cycle results that are improved here.","marker":"[3]"},{"why":"Original definition of $r(G,H,q)$ and the first bounds for the bipartite case that frame the problem.","marker":"[2]"},{"why":"McDiarmid's inequality is used to prove the concentration properties of the random linear hypergraphs $S_A$ and $S_B$.","marker":"[18]"},{"why":"Wilson's existence theorems for $(n,k-1,2)$-Steiner systems, the base hypergraphs from which $S_A$ and $S_B$ are sampled.","marker":"[22, 23, 24]"}],"fun_headline_variants":["Exact Ramsey rate for C6: 7/20 n","Sharp C6 bound: r(K_n,n,C6,3)=7/20 n","Bipartite C6 Ramsey constant: 0.35n","Improved bounds for even-cycle Ramsey","Lane-Morrison question resolved: tighter C2k bounds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The upper bound stands or falls with the black-box conflict-free matching theorem: if the hypergraph the paper constructs does not satisfy every one of its quantitative hypotheses, there is no guarantee that the required conflict-free perfect matching, and hence the coloring, exists.","fun_headline_variants_meta":{"raw":{"variants":["Exact Ramsey rate for C6: 7/20 n","Sharp C6 bound: r(K_n,n,C6,3)=7/20 n","Bipartite C6 Ramsey constant: 0.35n","Improved bounds for even-cycle Ramsey","Lane-Morrison question resolved: tighter C2k bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000327,"raw_usage":{"total_tokens":1798,"prompt_tokens":884,"completion_tokens":914,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":835}},"tokens_in":500,"tokens_out":914,"duration_ms":9746,"temperature":1.0,"reasoning_tokens":835,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:28:46.637364+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the parameter relation $p+q=k$ in Theorem 3.1 against the actual number of $P$ and $Q$ vertices in a generic $\\mathcal{H}_1$ edge; if the relation cannot be met, the upper-bound proof does not go through. A complementary check is to verify the inequalities (H1)--(H4), (C1)--(C3), and (D1)--(D4) numerically for a fixed small $k$, such as $k=4$, using $d$ as defined in equation (12).","supporting_citations":[{"cited_title":"Joos and D","cited_arxiv_id":null,"evidence_quote":"Introduced the conflict-free matching approach for generalized Ramsey upper bounds, including the asymptotic $r(K_{n,n}, C_4, 3)$ result that the paper extends."},{"cited_title":"Generalized Ramsey numbers via conflict-free hypergraph matchings","cited_arxiv_id":"2405.16653","evidence_quote":"Gives the previous upper bound with the integer floor and poses the question about removing it, which this paper answers."},{"cited_title":"Generalized Ramsey numbers of cycles, paths, and hypergraphs","cited_arxiv_id":"2405.15904","evidence_quote":"The authors' earlier joint paper with Heath and Zerbib gives the previous lower bound in (1) and related cycle results that are improved here."},{"cited_title":"Axenovich, Z","cited_arxiv_id":null,"evidence_quote":"Original definition of $r(G,H,q)$ and the first bounds for the bipartite case that frame the problem."},{"cited_title":"McDiarmid et al","cited_arxiv_id":null,"evidence_quote":"McDiarmid's inequality is used to prove the concentration properties of the random linear hypergraphs $S_A$ and $S_B$."}],"review_version":1}