{"id":"80ec0f13-8bcf-4916-ab33-690b4b0e8ee3","arxiv_id":"2605.26815","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The vertex-coloring coprime Ramsey number R_cop(k1,...,kc) equals the prime p indexed by sum(ki-1).","lead":"The paper proves that the mixed vertex-coloring coprime Ramsey number R_cop(k1,...,kc) equals the prime at position equal to the sum of (ki-1) over the colors. A smart generalist might read it to see an elementary bridge between prime numbers and exact Ramsey-type parameters in a specialized graph.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identified assumption about the clique is actually secure and does not represent a vulnerability. The proof sketch in the abstract is consistent with a correct argument once the handling of 1 in the lower bound is accounted for, which appears feasible as described.","tokens_in":1806,"tokens_out":230,"duration_ms":82244,"concrete_test":"Explicitly construct the avoiding coloring on [p_m - 1] for m = sum(ki-1) with specific small tuples (e.g., (2,2) and (3,2)) and verify that each color class has no ki pairwise coprime numbers.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern identified in the central claim. The upper bound follows directly from the clique size 1 + π(n) and pigeonhole on the color classes. The lower bound construction via prime bins accommodates the element 1 by designating one bin without primes and assigning 1 to it, ensuring no color class admits a ki-sized pairwise-coprime subset.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that the mixed vertex-coloring coprime Ramsey number satisfies R_cop(k_1,…,k_c)=p_{∑(k_i−1)}, where p_m is the m-th prime. The upper bound follows from the clique {1}∪{primes ≤ n} in the coprime graph G_n together with the pigeonhole principle on color classes; the matching lower bound is obtained from a prime-bin partition in which each composite is colored according to one of its prime factors. The same certificate framework yields an exact reduction Redge(k_1,…,k_c)=p_{R_cl(k_1,…,k_c)−1} for the edge-coloring variant, a support-disjointness generalization, and exact balanced two-color thresholds p_{2k−2} for all k≥2 via a deterministic prime-bin split that uses only the inequality 2p_m < p_{2m} < 3p_m.","tokens_in":1885,"tokens_out":421,"duration_ms":23965,"significance":"If the derivations hold, the result supplies a closed-form expression for an entire family of Ramsey-type numbers on the coprime graph, a rare occurrence in Ramsey theory. The elementary certificate construction, the direct reduction of the edge-coloring parameter to classical Ramsey numbers, and the provision of polynomial-time certificate extraction constitute concrete strengths that would be noted in any assessment of the work.","major_comments":[],"minor_comments":[{"comment":"The abstract states that the balanced two-color result holds for all k≥2 via the inequality 2p_m < p_{2m} < 3p_m; a short explicit verification of this inequality (or a reference to a standard bound) would make the deterministic split fully self-contained.","section":null},{"comment":"Notation: the symbols Rcop and Redge are introduced in the abstract but the manuscript should confirm that they are used consistently in all subsequent statements of the main theorems.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of the manuscript and the recommendation to accept.","responses":[],"tokens_in":1383,"tokens_out":36,"duration_ms":8730,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper pins down an exact formula for the mixed vertex-coloring coprime Ramsey number: it equals the prime whose index is the sum of the (ki minus one). The proof uses a straightforward clique argument for one direction and a prime-factor binning for the other.\n\nWhat works well is the identification of {1} union the primes as a large clique in the coprime graph. Pigeonhole on that clique immediately gives the upper bound once you color the vertices. The lower bound construction assigns each composite to one of its prime factors' bin and handles 1 separately, ensuring no color class has a large pairwise-coprime set. The same idea transfers to the edge-coloring variant, reducing it exactly to the classical Ramsey number shifted by one in the prime index. They also settle the balanced diagonal case for two colors using a simple split that relies only on a weak prime inequality.\n\nThe work is new in giving this closed form and the certificate reductions. The math is direct, with no hidden parameters or circular reductions. The citation pattern is light and focused on the relevant background.\n\nThe soft spots are minor. The multicolor balanced result is eventual rather than for all n, but that's clearly stated. For the main claim the bounds match exactly, so the formula holds if the constructions are correct, which the sketch supports.\n\nThis paper is aimed at people studying Ramsey-type problems on integer graphs or exact values in combinatorial number theory. A reader looking for closed forms in this setting will find something concrete to use or extend. It is worth a serious referee because the claim is sharp and the argument is elementary enough to review thoroughly.","headline":"The paper delivers an exact closed-form for the vertex-coprime Ramsey number as the right prime, with matching elementary bounds that check out on the given sketch.","tokens_in":2381,"tokens_out":410,"would_cite":true,"duration_ms":40935,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The mixed vertex-coloring coprime Ramsey number R_cop(k_1,…,k_c) equals the prime whose index is the sum of (k_i-1).","keywords":["coprime Ramsey numbers","vertex-coloring","prime certificates","coprime graphs","Ramsey theory","exact formulas","prime partitions"],"falsifier":"An explicit c-coloring of the coprime graph on the integers from 1 to p_m minus 1, where m equals the sum of (k_i minus 1), that avoids a monochromatic coprime clique of size k_i in color i for each i.","tokens_in":2702,"feed_emoji":"🔢","tokens_out":757,"duration_ms":52122,"temperature":0.7,"pith_summary":"The paper establishes an exact closed-form expression for the mixed vertex-coloring coprime Ramsey numbers on the coprime graph G_n. It shows these numbers equal the prime at position equal to the sum over i of (k_i minus one). A sympathetic reader would care because the result replaces the usual search for Ramsey thresholds with a direct reference to the sequence of primes. The upper bound follows from the pigeonhole principle applied to a large clique, while the lower bound follows from a partition that routes each composite number into one of its prime factors.","feed_headline":"Coprime Ramsey numbers equal primes at summed index","feed_subtitle":"R_cop(k1 to kc) equals the prime whose position is the sum of each (ki minus one), via elementary clique and bin arguments.","key_machinery":"The clique formed by 1 together with all primes up to n in the coprime graph G_n, which supplies the pigeonhole upper bound, together with the prime-bin partition that supplies the matching lower bound.","core_discovery":"We prove that the mixed vertex-coloring coprime Ramsey number satisfies R_cop(k_1,…,k_c)=p_{∑_{i=1}^c(k_i-1)}, where p_m is the m-th prime. The proof is elementary: the prime clique {1}∪{p≤n:p prime} gives the upper bound by pigeonhole, while a prime-bin partition gives the matching lower bound by coloring each composite with a bin containing one of its prime divisors. The same certificate viewpoint yields an exact reduction of the edge-coloring variant to classical Ramsey numbers: R_edge(k_1,…,k_c)=p_{R_cl(k_1,…,k_c)-1}.","pith_inferences":["The formula directly ties the growth of these Ramsey numbers to the distribution of primes.","The certificate method may transfer to other Ramsey parameters defined on graphs whose edges or vertices are governed by divisibility or coprimality conditions."],"forward_implications":["The edge-coloring parameter satisfies the exact reduction R_edge(k_1,…,k_c)=p_{R_cl(k_1,…,k_c)-1}.","The balanced two-color diagonal threshold equals the unrestricted threshold p_{2k-2} for every k at least 2.","For any fixed number of colors c, multicolor balanced certificates exist for all sufficiently large parameters via a Hall argument combined with the Selberg-Delange estimate.","A polynomial-time primitive exists for extracting the certificates from the same clique-label construction."],"fun_headline_variants":["Coprime Ramsey equals prime at summed index","Coprime Ramsey ties to prime via index sum","Sum index prime defines coprime Ramsey","Vertex coprime Ramsey is prime at index sum","Coprime Ramsey formula uses summed prime index"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The set consisting of 1 together with the primes up to the relevant prime forms a clique in the coprime graph G_n.","fun_headline_variants_meta":{"raw":{"variants":["Coprime Ramsey equals prime at summed index","Coprime Ramsey ties to prime via index sum","Sum index prime defines coprime Ramsey","Vertex coprime Ramsey is prime at index sum","Coprime Ramsey formula uses summed prime index"]},"model":"grok-4.3","cost_usd":0.008782,"raw_usage":{"total_tokens":4022,"prompt_tokens":803,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":87824500,"prompt_tokens_details":{"text_tokens":803,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3154,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":803,"tokens_out":65,"duration_ms":35816,"temperature":1.0,"reasoning_tokens":3154,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T17:23:08.174014+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit c-coloring of the coprime graph on the integers from 1 to p_m minus 1, where m equals the sum of (k_i minus 1), that avoids a monochromatic coprime clique of size k_i in color i for each i.","supporting_citations":[],"review_version":1}