{"id":"2f540ad6-dc80-408c-a5f6-371bbda6b226","arxiv_id":"2507.19456","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For every fixed t and k, r_odd(K_{n,n}, K_{2,t}) = n/t + o(n) and r_odd(K^{(k)}_{n,...,n}, K_{1,...,1,2,2}) = n/2 + o(n).","lead":"This paper proves that the odd Ramsey number for the complete bipartite graph K_{n,n} against K_{2,t} is n/t plus lower-order terms, and extends the result to complete k-partite hypergraphs against K_{1,...,1,2,2}. A generalist should care because it is the first hypergraph result in odd Ramsey theory and demonstrates a new conflict-free matching technique.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Conflict system C in Definition 5 is not bounded as claimed: inclusion-minimal irreducible K2,r conflicts with r>t (indeed arbitrarily large |C|) exist, so the r≤t assumption in §3.1.1 and |C|≤2t are false as stated.","rationale":"The reader's weakest assumption identifies exactly the load-bearing gap: the proof requires every inclusion-minimal irreducible conflict to have an irreducible bad copy with r≤t, and the degree bookkeeping in §3.1.1 depends on that restriction. The cycle construction above shows the restriction is not just unproved but false for the conflict system as defined: there are irreducible bad K2,r conflicts with r>t and with |C| arbitrarily large, so condition (C1) fails. This is an internal inconsistency in the manuscript's application of Theorem 3, not a disagreement with external consensus. The underlying asymptotic results are plausible and the gap is localized: one can likely redefine C to include only conflicts arising from bad copies with r≤t, since any bad K2,t decomposes into such conflicts, and then re-run the degree counts. Because the fix is concrete but the written proof is not correct as it stands, the reader's CONDITIONAL verdict remains appropriate.","tokens_in":24845,"tokens_out":27273,"duration_ms":274577,"concrete_test":"For t=3, instantiate the eight tiles described above and verify (i) pairwise vertex intersections in Kn,n are at most one, so the tiles form a matching in H1 after adding disjoint auxiliary vertices; (ii) each color occurs twice, so K2,8 is bad; (iii) the color incidence graph is an 8-cycle, so the tiles form an inclusion-minimal irreducible conflict of size 8. This directly refutes |C|≤2t and r≤t for Definition 5. Then check whether redefining C to contain only irreducible conflicts with r≤t satisfies Claims 7–8 and whether the degree counts in §3.1.1 then go through.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 5 defines C as the set of all inclusion-minimal irreducible conflicts, for arbitrary bad copies K2,r with r≥1. Section 3.1.1 then asserts that for each C∈C one may fix an irreducible bad copy GC with 2≤r≤t, and concludes from this that 3≤|C|≤2t. This assertion is not merely unproved; it is false for the conflict system as defined. For t=3, take leaves y1,...,y8 and X-tiles P1={y1,y2}, P2={y3,y4}, P3={y5,y6}, P4={y7,y8} (each tile contains x1 and those two leaves) and Y-tiles Q1={y1,y3}, Q2={y4,y5}, Q3={y6,y7}, Q4={y8,y2} (each tile contains x2 and those two leaves). Give the eight tiles eight distinct colors; every color occurs exactly twice, so K2,8 is bad. The incidence graph between the Pi and Qi is an 8-cycle, so no proper subset of leaves has all color counts even; hence no proper bad subcopy exists and the eight-tile set is an inclusion-minimal irreducible conflict. For sufficiently large n the tiles can be completed to K4,4-minus-a-perfect-matching tiles and form a matching in H1, since any two share at most one graph vertex. Thus C has size 8 for t=3 (and, using longer cycles, arbitrarily large size), contradicting (C1) and the claimed r≤t. Claims 9 and 10 rely throughout on r≤t; Claim 10 even writes 'GC∼=K2,t'. The proof can likely be repaired by defining C over irreducible bad copies with r≤t only, because a bad K2,t decomposes into such conflicts, but the manuscript applies Theorem 3 to a conflict system whose boundedness is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies odd Ramsey numbers, the minimum number of colors in an edge-coloring of a host hypergraph such that every copy of a target subhypergraph has some color class of odd size. The main results are Theorem 1, r_odd(K_{n,n}, K_{2,t}) = n/t + o(n) for all t >= 2, and Theorem 2, r_odd(K^{(k)}_{n,...,n}, K_{1,...,1,2,2}) = n/2 + o(n) for all k >= 2. The lower bounds are simple averaging arguments. The upper bounds use the Tripartite Matching Theorem of Joos, Mubayi, and Smith: the authors construct hypergraphs H1 and H2 whose matchings encode colorings of the host graph/hypergraph, and then define conflict systems C and D that are supposed to forbid all bad copies of the target. The paper is organized as a bipartite-graph proof in Section 3 and a hypergraph proof in Section 4.","tokens_in":25242,"tokens_out":23595,"duration_ms":243395,"significance":"If the proofs were correct, these would be attractive tight asymptotic results. Theorem 1 generalizes the known K_{2,2} case, and Theorem 2 would be the first odd Ramsey result for hypergraphs. The lower-bound arguments are clean, and the intended use of the Tripartite Matching Theorem is appropriate. The paper gives explicit constructions rather than abstract existence arguments, and the overall strategy is well matched to the problem. However, the manuscript currently contains load-bearing gaps in the definition and boundedness of the conflict systems, both in the graph case and, more seriously, in the hypergraph case.","major_comments":[{"comment":"The claim that every conflict in C satisfies 3 <= |C| <= 2t, and the accompanying assertion that for each C in C one may fix an irreducible bad copy G_C of K_{2,r} with 2 <= r <= t, are not justified and are in fact false for the conflict system as defined. For t = 3, take leaves y_1,...,y_8 and eight tiles: P_1 = {x_1, y_1, y_2}, P_2 = {x_1, y_3, y_4}, P_3 = {x_1, y_5, y_6}, P_4 = {x_1, y_7, y_8}, and Q_1 = {x_2, y_1, y_3}, Q_2 = {x_2, y_4, y_5}, Q_3 = {x_2, y_6, y_7}, Q_4 = {x_2, y_8, y_2}. Giving the eight tiles eight distinct colors makes K_{2,8} bad, since every color appears on exactly two edges. The incidence graph between the P_i and Q_j is an 8-cycle, so no proper subset of leaves has all color counts even; hence this bad K_{2,8} is irreducible and no smaller bad subcopy exists. For sufficiently large n the tiles can be completed to K_{4,4}-minus-perfect-matching tiles that form a matching in H1, because any two of them share at most one graph vertex and their colors are distinct. Thus C contains an inclusion-minimal irreducible conflict of size 8, contradicting (C1); using longer cycles gives arbitrarily large conflicts. Consequently the proof of (C1) and the choice of G_C with r <= t in Section 3.1.1 are invalid, and Claims 9 and 10, which rely on that choice, do not establish condition (C3) for the C that is actually defined. The proof can likely be repaired by redefining C to consist only of inclusion-minimal irreducible conflicts whose witness bad copy has r <= t, but this must be proved and the degree bounds must be re-checked for that system.","section":"Section 3.1 / Definition 5 / Section 3.1.1"},{"comment":"The conflict system D has the same defect as C. The text says that every conflict in D has at least two tiles from H2 and no more than 2t tiles, but Definition 12 does not restrict the parameter r of the irreducible bad K_{2,r} used as a witness. The same construction as in the previous comment can be repeated with H2 tiles: give each of the 16 edges of a K_{2,8} its own H2 tile and pair the colors so that each color appears twice; the resulting set is an inclusion-minimal irreducible conflict with arbitrarily many tiles when extended cyclically. Thus property (D1) is not established, and the degree counts in Sections 3.2.1-3.2.3, which explicitly assume 2 <= r <= t, do not apply to the D defined in the manuscript. The repair suggested for C should also be applied to D.","section":"Section 3.2 / Definition 12"},{"comment":"The proof of Theorem 2 does not handle a bad copy of K_{1,...,1,2,2} contained in a single H1 tile. A tile e_{S,i} contains every transversal of S, i.e. every k-tuple with one vertex from each part of S. If S has two vertices in each of two parts and one vertex in each of the remaining k-2 parts, then the four corresponding hyperedges form a copy of K_{1,...,1,2,2} and all four receive color i from that tile. Such a copy has no odd color class. However, the conflict system C in Section 4.1 consists only of conflicts of size 3 or 4 that use two distinct colors; there is no conflict for the one-tile case. Thus a P-perfect matching avoiding C union D can still contain a tile that by itself creates a bad copy of the target. The statement that 'the graph in Fig. 4 is colored in the only way that is allowed by a matching of H1' appears to overlook this possibility. The construction of H1 or the definition of the conflicts must be changed, for example by deleting from each S a suitable set of hyperedges so that no single tile contains a bad copy, and the degree computations would then need to be redone.","section":"Section 4.1 / Figure 4"}],"minor_comments":[{"comment":"Claim 10 writes 'GC ~= K_{2,t}', but the discussion at the start of Section 3.1.1 only guarantees 2 <= r <= t. The leaf-counting argument should be written for K_{2,r} with general r <= t, and the final equality L_{j-j'} = 0 should refer to r, not to t.","section":"Section 3.1.1, Claim 10"},{"comment":"The displayed estimate d^epsilon = O(n^{(k^2+1)/(k^2+2)}) is missing a factor 1/2: from d = n^{(k^2+1)/2} and epsilon = 1/(k^2+2) one obtains d^epsilon = n^{(k^2+1)/(2(k^2+2))}. The subsequent inequality is still true, but the formula should be corrected.","section":"Section 4, after Claim 18"},{"comment":"The notation binom{n}{delta} is used for n^delta in several places. Since delta is not an integer, this is formally incorrect; use n^delta or lceil n^delta rceil consistently.","section":"Sections 3.2.1 and 4.2"}],"recommendation":"major_revision","confidential_remarks":"The Section 4 issue with one-tile bad copies is serious and may require a substantial reworking of the construction, not just a local fix to the conflict system. The Section 3 issue with unbounded irreducible conflicts appears repairable by restricting the conflict system to r <= t, but the authors must prove that the restricted system suffices and then verify all conditions of the Tripartite Matching Theorem for that system. If the hypergraph construction cannot be repaired, the paper's Theorem 2 would be unsupported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this is a useful paper with a genuine hole in the main graph-theoretic upper bound. Theorems 1 and 2 are new and probably true, but as written the conflict system for Theorem 1 is not bounded as claimed.\n\nWhat is good: the lower bound for Theorem 1 is a clean averaging argument, the hypergraph theorem is the first of its kind, and the paper applies the Tripartite Matching Theorem in a serious way. The exposition is clear and the hypergraph section has explicit, small conflicts with degree counts that mostly look plausible. This is a real contribution to the odd-Ramsey program.\n\nThe soft spot: Definition 5 defines C as all inclusion-minimal irreducible conflicts for arbitrary bad copies K_{2,r}, and Section 3.1.1 asserts that one can fix an irreducible bad copy G_C with 2 ≤ r ≤ t, hence 3 ≤ |C| ≤ 2t. That assertion is false. For t = 3, take leaves y1,...,y8 and tiles P_i = {x1, y_{2i-1}, y_{2i}} and Q_j = {x2, ...} arranged so the incidence graph is an 8-cycle. Give the eight tiles eight distinct colors. Every color appears exactly twice, so K_{2,8} is bad, and because the incidence graph is a single cycle, no proper subset of leaves has all color counts even. So this is an inclusion-minimal irreducible conflict of size 8, contradicting |C| ≤ 2t = 6. Longer cycles give arbitrarily large |C|. Hence C is not (d, ℓ, ε)-bounded and Theorem 3 cannot be applied to this system. Claims 9 and 10, and the degree bounds (C2) and (C3), all build on the r ≤ t assumption.\n\nIn proportion: this is a fixable flaw, not a sign that the theorems are wrong. The natural repair is to restrict C and D to irreducible bad copies with r ≤ t, since the target is K_{2,t} and any decomposition of it involves only r_i ≤ t. Claims 6–8 survive under that restriction, and the counting in Section 3.1.1 would then have a valid range. But the repair has to be written and the boundedness rechecked; the proof as submitted is conditional.\n\nWho this is for: anyone working on odd Ramsey numbers or conflict-free hypergraph matchings. It deserves a serious referee, because the results matter and the argument is close. My recommendation: send it to review, but the referee should require the conflict-system fix before acceptance. I would not cite it in its current form.","headline":"A likely-true pair of theorems, but the main boundedness claim for the conflict system in Theorem 1 is false as stated and needs a repair.","tokens_in":25791,"tokens_out":5742,"would_cite":false,"duration_ms":54195,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C55","05C15","05C65","05D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Odd-Ramsey count for K_{2,t} is n/t asymptotically","keywords":["odd Ramsey numbers","complete bipartite graphs","multipartite hypergraphs","edge colorings","odd color class","conflict-free hypergraph matchings","asymptotic Ramsey theory","K_{2,t}"],"falsifier":"Exhibit an irreducible bad coloring of $K_{2,t+1}$ built from the paper's tiles inside a matching of $H_1$, with every color class even, that cannot be split into two nonempty bad subcopies on the same pair of size-2 vertices. Such a configuration would violate the claimed bound $|C|\\le 2t$ and the codegree calculation underlying condition (C3).","tokens_in":24642,"feed_emoji":"🎨","tokens_out":9146,"duration_ms":83002,"temperature":0.7,"pith_summary":"This paper proves exact asymptotic formulas for two odd Ramsey numbers, the minimum number of edge colors needed so that every copy of a given subgraph uses some color an odd number of times. For the complete bipartite graph $K_{n,n}$ and the target $K_{2,t}$, the answer is $n/t+o(n)$ for every $t\\ge 2$. For the complete $k$-partite $k$-uniform hypergraph $\\mathcal{K}^{(k)}_{n,\\ldots,n}$ and the target $\\mathcal{K}_{1,\\ldots,1,2,2}$, the answer is $n/2+o(n)$ for every $k\\ge 2$, which is the first odd Ramsey result for hypergraphs. The lower bounds are obtained by a pair-counting argument, while the upper bounds are obtained by constructing colorings through a tripartite conflict-free hypergraph matching theorem.","feed_headline":"Odd-Ramsey count for K_{2,t} is n/t asymptotically","feed_subtitle":"Matching arguments also give the first hypergraph odd-Ramsey result: n/2 colors.","key_machinery":"The load-bearing mechanism is the Tripartite Matching Theorem, a version of the conflict-free hypergraph matching method that produces a perfect matching of an auxiliary hypergraph while avoiding two prescribed families of forbidden configurations, called conflicts. In the graph proof, the edges of $H_1$ are monochromatic tiles, copies of $K_{t+1,t+1}$ with a perfect matching removed, and the edges of $H_2$ are single colored edges of $K_{n,n}$; a matching in $H=H_1\\cup H_2$ corresponds to a well-defined edge-coloring. The conflict systems $C$ and $D$ collect inclusion-minimal irreducible configurations whose colored edges would contain a bad $K_{2,t}$, meaning a copy with no odd color class. The hypergraph proof uses the same architecture, with transversals of $K^{(k)}_{k+1,\\ldots,k+1}$ as tiles and conflicts corresponding to bad copies of $\\mathcal{K}_{1,\\ldots,1,2,2}$. The whole argument works because the auxiliary hypergraphs and conflict systems satisfy the degree and codegree bounds required by the matching theorem.","core_discovery":"On its own terms, the paper establishes that $r_{\\mathrm{odd}}(K_{n,n}, K_{2,t}) = n/t + o(n)$ for all $t\\ge 2$ and that $r_{\\mathrm{odd}}(\\mathcal{K}^{(k)}_{n,\\ldots,n}, \\mathcal{K}_{1,\\ldots,1,2,2}) = n/2 + o(n)$ for all $k\\ge 2$. The first statement generalizes the previously known case $t=2$; the second is the first asymptotically tight odd Ramsey result for hypergraphs. The lower-bound proof shows that fewer than $n/t$ (respectively $n/2$) colors force, by Cauchy–Schwarz and pigeonhole, a bad copy in which every color class has even size. The upper-bound proof encodes a coloring of most of the host graph as a matching in an auxiliary hypergraph and defines conflict systems that forbid exactly those configurations that would leave a target copy without an odd color class; a one-step matching theorem removes the separate recoloring step used by earlier arguments.","pith_inferences":["A natural next step is to apply the same one-step matching recipe to targets such as $K_{2,t}$ in other bipartite hosts or to hypergraph targets with three distinguished parts; the likely bottleneck is classifying irreducible bad colorings rather than the matching theorem itself.","The proof does not make the $o(n)$ term explicit. A sharper analysis of the leaf-elimination counting in the star case could plausibly yield a polynomial error bound such as $O(n^{1-\\varepsilon})$, which could be checked computationally for small $n$.","The overall structure suggests a transfer principle: once a target's bad colorings are understood, the odd Ramsey number against a complete multipartite host may be governed by the largest side of the target, as $1/t$ and $1/2$ appear here."],"forward_implications":["For every fixed $t\\ge 2$, the asymptotic value $n/t$ determines the odd Ramsey number of $K_{2,t}$ inside $K_{n,n}$ up to an $o(n)$ error term.","The hypergraph statement makes $n/2+o(n)$ the first known asymptotic odd Ramsey number for uniform hypergraphs, and it holds simultaneously for every uniformity $k\\ge 2$.","Through the graph-code inequality cited in the introduction, these upper bounds imply lower bounds on the maximum density of graph codes that avoid the corresponding subgraphs.","The lower-bound counting argument identifies the constant $1/t$ as forced by pair counting: with fewer colors, two same-colored edges incident to one vertex necessarily produce a bad $K_{2,t}$."],"supporting_citations":[{"why":"The base result $r_{\\mathrm{odd}}(K_{n,n},K_{2,2})=\\frac12 n+o(n)$ that Theorem 1 generalizes, together with the conflict-free matching coloring technique.","marker":"[9]"},{"why":"Supplies the Tripartite Matching Theorem, the one-step conflict-free matching tool used in both upper-bound proofs.","marker":"[21]"},{"why":"Introduces the conflict-free hypergraph matching method and the boundedness conditions that the conflict systems $C$ and $D$ are checked against.","marker":"[18]"}],"fun_headline_variants":["Odd Ramsey: n/t for K_{2,t}, n/2 for hypergraphs","First hypergraph odd Ramsey number: n/2 asymptotically","Odd Ramsey for K_{n,n} vs K_{2,t}: n/t + o(n) colors","n/t colors force odd K_{2,t} in K_{n,n}"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every minimal irreducible bad configuration can be witnessed by a copy of $K_{2,r}$ with $r\\le t$; if an irreducible bad copy with $r>t$ can occur in a matching, the bound $|C|\\le 2t$ and the codegree estimates in Section 3.1.1 would need additional justification.","fun_headline_variants_meta":{"raw":{"variants":["Odd Ramsey: n/t for K_{2,t}, n/2 for hypergraphs","First hypergraph odd Ramsey number: n/2 asymptotically","Odd Ramsey for K_{n,n} vs K_{2,t}: n/t + o(n) colors","n/t colors force odd K_{2,t} in K_{n,n}"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000526,"raw_usage":{"total_tokens":2533,"prompt_tokens":932,"completion_tokens":1601,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":1513}},"tokens_in":548,"tokens_out":1601,"duration_ms":12760,"temperature":1.0,"reasoning_tokens":1513,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:55:13.406926+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit an irreducible bad coloring of $K_{2,t+1}$ built from the paper's tiles inside a matching of $H_1$, with every color class even, that cannot be split into two nonempty bad subcopies on the same pair of size-2 vertices. Such a configuration would violate the claimed bound $|C|\\le 2t$ and the codegree calculation underlying condition (C3).","supporting_citations":[{"cited_title":"Glock, F","cited_arxiv_id":null,"evidence_quote":"Introduces the conflict-free hypergraph matching method and the boundedness conditions that the conflict systems $C$ and $D$ are checked against."}],"review_version":2}