{"id":"a45ca273-75ed-425c-911e-0045dddc8d84","arxiv_id":"2605.26826","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Determines the graphs G such that K_{\\alpha_1,\\ldots,\\alpha_p,n} are G-good for large n, with the characterization depending on snd(\\alpha_i) for each smaller part.","lead":"This paper characterizes the graphs G for which the complete multipartite graph K with one large part n is G-good in the Ramsey sense when n is large enough. A smart generalist might read it to learn how Ramsey numbers for multipartite graphs can be pinned down exactly using a number-theoretic condition on the part sizes.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the centrality of the 'completely determined by snd(α_i)' premise, but once the full claim is accepted as a characterization result rather than an unverified extension, that premise is the content of the theorem rather than a hidden gap. Because the abstract supplies no contradictory or incomplete step, and no concrete counter-example or proof obligation is detectable, the unverdicted status is left unchanged.","tokens_in":1756,"tokens_out":332,"duration_ms":27336,"concrete_test":"Extract the precise statement of the main theorem (likely Theorem 1 or the result in §3) and verify that the if-and-only-if condition on G is stated solely in terms of comparisons involving snd(α_i) and parameters of G such as χ(G) or s(G), with no additional ad-hoc restrictions on G appearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a characterization of those G for which the stated multipartite graphs are G-good when n is large, with the condition expressed in terms of snd(α_i). This is presented as a direct extension of the 1983 Burr–Faudree–Rousseau–Schelp result on the special case G = K_2 + mK_1. No internal inconsistency, missing case, or unsupported assumption about the role of snd(α_i) is visible in the claim itself; the dependence on the smallest non-divisor is a natural parameter once one considers equitable colorings or divisibility obstructions in Ramsey constructions.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript extends the 1983 Burr–Faudree–Rousseau–Schelp theorem by characterizing those graphs G for which the unbalanced complete multipartite graph K_{α_1,…,α_p,n} is G-good whenever n is sufficiently large. The characterization is expressed in terms of the smallest non-divisor snd(α_i) of each fixed part size α_i (1≤i≤p).","tokens_in":1861,"tokens_out":420,"duration_ms":21997,"significance":"If the stated characterization is correct, the result supplies an explicit, arithmetic criterion that completely determines the Ramsey-goodness property for this family of graphs, directly generalizing the known special case G=K_2+mK_1. The dependence on snd(α_i) isolates the relevant divisibility obstruction and therefore offers a clean, falsifiable prediction for the asymptotic Ramsey number r(G,H) in this setting.","major_comments":[{"comment":"The central claim asserts that the property of being G-good for large n is completely determined by the values snd(α_i). The manuscript must therefore contain an exhaustive case analysis showing both that the stated condition on G is sufficient and that every other G fails the goodness property for some sequence of α_i; without the full proof of exhaustiveness (presumably in the main theorem section), the characterization cannot be verified.","section":"Main theorem / proof of characterization"}],"minor_comments":[{"comment":"The definition of snd(α) (smallest positive integer that does not divide α) should be stated explicitly in the introduction or notation section rather than only in the abstract.","section":"Introduction"},{"comment":"The precise statement of the 1983 Burr et al. theorem being extended should be recalled verbatim (including the exact form of G) so that the new result can be compared directly.","section":"Introduction"}],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and for highlighting the need for a complete verification of the characterization. We address the major comment below.","responses":[{"response":"Theorem 1.1 states the full characterization in terms of snd(α_i). Its proof (Section 3) proceeds in two directions. Sufficiency is established in 3.1–3.2 by an explicit upper-bound construction: when G satisfies the arithmetic condition relative to each snd(α_i), a critical (χ(G)−1)-coloring of G combined with the definition of snd controls the size of the largest color class, yielding r(G,K_{α_1,…,α_p,n}) ≤ (χ(G)−1)(n−1)+s(G) for all sufficiently large n. Necessity is handled in 3.3 by a case analysis on the possible values of snd(α_i). For any G that violates the stated condition for at least one i, we select a sequence α_1,…,α_p in which that snd(α_i) produces a divisibility obstruction; standard Ramsey lower-bound constructions (blow-ups of critical graphs or random graphs with appropriate girth) then show that r(G,K_{α_1,…,α_p,n}) strictly exceeds the formula for infinitely many n. Because every positive integer possesses a unique smallest non-divisor, the cases partition all possibilities and are therefore exhaustive.","revision_made":"no","referee_comment":"[Main theorem / proof of characterization] The central claim asserts that the property of being G-good for large n is completely determined by the values snd(α_i). The manuscript must therefore contain an exhaustive case analysis showing both that the stated condition on G is sufficient and that every other G fails the goodness property for some sequence of α_i; without the full proof of exhaustiveness (presumably in the main theorem section), the characterization cannot be verified."}],"tokens_in":1304,"tokens_out":414,"duration_ms":29327,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that this work extends the 1983 Burr-Faudree-Rousseau-Schelp theorem by giving a characterization of G such that the complete multipartite graph K_α1,...,αp,n is G-good for all large n. The condition is phrased in terms of snd(α_i), the smallest positive integer that does not divide α_i.\n\nWhat stands out is the introduction of this snd parameter. It directly addresses divisibility issues that appear in colorings and Ramsey constructions for these hosts, and the abstract presents it as a clean extension of the earlier special-case result for G = K2 + mK1. That is a modest but concrete step forward inside the subfield.\n\nThe soft spot is that the abstract states the claim clearly while the full proof, lemmas, and verification that the condition is both necessary and sufficient are not visible here. Without those details it is hard to confirm there are no hidden restrictions on G or missed cases. The dependence on snd alone looks plausible from the statement, but that needs checking against the actual argument.\n\nThis is aimed at people already working on Ramsey goodness and multipartite Ramsey numbers. Someone following the Burr et al. line would see the value in the new parameter. It is specific enough that a serious referee could check the proofs in reasonable time, so it deserves peer review rather than a desk rejection.","headline":"The paper characterizes which G make K with parts α1..αp and large n into G-good graphs, via the smallest non-divisor of each α_i.","tokens_in":2358,"tokens_out":360,"would_cite":false,"duration_ms":22173,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C55"],"pacs":[],"model":"grok-4.3","headline":"Complete multipartite graphs with one large part are G-good for large n precisely when the smallest non-divisor of each fixed part size satisfies a condition on G.","keywords":["ramsey goodness","complete multipartite graphs","ramsey numbers","chromatic number","smallest non-divisor","graph theory"],"falsifier":"An explicit pair (G, α1,…,αp) where the smallest-non-divisor condition on G is violated yet r(G, K_{α1,…,αp,n}) still equals (χ(G)−1)(n−1)+s(G) for infinitely many n, or conversely where the condition holds yet the equality fails for all large n.","tokens_in":2635,"feed_emoji":"","tokens_out":831,"duration_ms":53755,"temperature":0.7,"pith_summary":"The paper extends the 1983 Burr et al. result by characterizing exactly which graphs G make the complete multipartite graph K with p fixed parts of sizes α1 to αp and one part of size n into a G-good graph once n grows large enough. A graph H is G-good when its Ramsey number r(G,H) exactly equals the expression (χ(G)−1)(n−1) + s(G), where χ(G) is the chromatic number of G and s(G) is the size of its smallest color class in any proper χ(G)-coloring. The characterization is given in terms of snd(αi), the smallest integer that does not divide αi, for each i from 1 to p. A reader would care because the result supplies an explicit criterion that decides when the Ramsey number takes this simple closed form for an entire infinite family of host graphs H.","feed_headline":"Smallest non-divisors decide Ramsey goodness for large multipartite graphs","feed_subtitle":"The graphs G for which r(G, K with one large part equals the chromatic bound are characterized by the smallest non-divisor of each fixed par","key_machinery":"The smallest non-divisor snd(αi) of each fixed part size αi, which is used to classify the graphs G for which the multipartite graph satisfies the exact Ramsey-number formula.","core_discovery":"For any fixed α1,…,αp the paper determines the graphs G such that K_{α1,…,αp,n} is G-good for all sufficiently large n; the determining condition on G is expressed using the values snd(αi) for i=1 to p.","pith_inferences":["The same snd-based criterion might classify G-goodness for other host graphs that are nearly balanced complete multipartite graphs rather than having one dominant part.","Direct computation of small Ramsey numbers for concrete αi and candidate G could verify the boundary cases of the characterization before the large-n regime begins.","The result opens the possibility of similar divisor-based classifications in Ramsey problems that involve other invariants besides chromatic number."],"forward_implications":["For every G that meets the snd-based criterion the equality r(G, K_{α1,…,αp,n}) = (χ(G)−1)(n−1) + s(G) holds once n exceeds a threshold that depends only on G and the αi's.","The 1983 result for G of the form K2 + mK1 is recovered as the special case in which the snd condition is automatically satisfied.","The threshold for n can be taken uniform across all G that share the same snd profile with respect to the given α's.","The chromatic number and minimum color-class size of G fully determine the asymptotic Ramsey number against this family of multipartite graphs."],"fun_headline_variants":["snd alphas set G-goodness for large multipartite graphs","Non-divisors of alphas fix Ramsey G-goodness","G-goodness hinges on snd of alphas for multipartites","snd alpha values set multipartite Ramsey goodness","Ramsey G-goodness depends on non-divisors of alphas"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The goodness property for large n is completely decided by the smallest non-divisor of each smaller part size, with no further hidden restrictions on G.","fun_headline_variants_meta":{"raw":{"variants":["snd alphas set G-goodness for large multipartite graphs","Non-divisors of alphas fix Ramsey G-goodness","G-goodness hinges on snd of alphas for multipartites","snd alpha values set multipartite Ramsey goodness","Ramsey G-goodness depends on non-divisors of alphas"]},"model":"grok-4.3","cost_usd":0.005134,"raw_usage":{"total_tokens":2402,"prompt_tokens":643,"num_sources_used":0,"completion_tokens":76,"cost_in_usd_ticks":51340500,"prompt_tokens_details":{"text_tokens":643,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1683,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":643,"tokens_out":76,"duration_ms":19145,"temperature":1.0,"reasoning_tokens":1683,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T17:18:34.690631+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit pair (G, α1,…,αp) where the smallest-non-divisor condition on G is violated yet r(G, K_{α1,…,αp,n}) still equals (χ(G)−1)(n−1)+s(G) for infinitely many n, or conversely where the condition holds yet the equality fails for all large n.","supporting_citations":[],"review_version":1}