{"id":"bc3b0c53-13ff-4671-bd35-f7ba4b7ebcb2","arxiv_id":"2604.16188","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Authors compute new small two-color ordered and cyclic Ramsey numbers for monotone paths, cycles, stars, complete graphs and nested matchings via SAT solving, determine closed forms for several pairs of graph classes, obtain bounds, apply reinforcement learning for lower bounds, and introduce permut","lead":"The paper uses a SAT solver to compute new small values for ordered Ramsey numbers of paths, cycles, stars and other graphs, and introduces cyclic Ramsey numbers as a relaxed variant along with a group-theoretic unification via permutational Ramsey numbers. A smart generalist might read it to understand how computational tools are generating exact data and new definitions in ordered variants of classical Ramsey theory.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"SAT encoding correctness for ordered/cyclic conditions is the least-secured step before pattern extrapolation to closed forms.","rationale":"The reader's weakest assumption already isolates the exact point where the argument is most fragile. Because the paper's novelty lies in the new closed forms and bounds derived from the tables, any undetected encoding error propagates directly to those claims. The proposed re-encoding check is a minimal, falsifiable test that would either confirm the foundation or force retraction of the general results. No other internal inconsistency is visible from the abstract and claim structure.","tokens_in":1920,"tokens_out":366,"duration_ms":20099,"concrete_test":"Take the smallest non-trivial instance the paper claims to have computed (e.g., R_ord(P_3,P_4) or the first cyclic case listed in Table 1). Re-encode it from the formal definition in §2 using an independent SAT front-end (e.g., PySAT or direct DIMACS) and solve with a second solver (Glucose or Z3); if the satisfiability outcome or witness differs from the paper's reported value, the encoding is faulty.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"All headline determinations of entire classes (e.g., R_ord(P_m, C_n) or similar pairs) and the reported small values rest on Kissat runs whose encoding of the order-preserving embedding (for ordered) or rotational invariance (for cyclic) is not independently verified in the supplied text. A single omitted clause or incorrect variable mapping in the CNF would silently produce wrong tables; the subsequent 'structural patterns' observed in those tables would then be artifacts. No cross-check against known classical Ramsey numbers, no hand-verified small instances, and no machine-checked encoding appear to be supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper defines the ordered Ramsey number R_ord(H1,...,Hk) as the smallest n such that every k-edge-coloring of the ordered complete graph on {0,...,n-1} contains a monochromatic ordered copy of each Hj in color j. It applies the Kissat SAT solver to compute new small two-color ordered Ramsey numbers for classes including monotone paths, monotone cycles, alternating paths, stars, complete graphs, and nested matchings. The paper introduces cyclic Ramsey numbers R_cyc as a relaxation of the ordered variant and computes small values for the same classes. By identifying structural patterns in the computed tables, it claims closed-form determinations for all ordered or cyclic Ramsey numbers in several pairs of graph classes, provides bounds when one argument is a connected graph and the other a monotone path or cycle, explores reinforcement learning via the RLGT framework for lower bounds, and introduces permutational Ramsey numbers to unify standard, ordered, and cyclic variants in a group-theoretic setting.","tokens_in":2094,"tokens_out":665,"duration_ms":32571,"significance":"If the SAT computations are correct, the work supplies new exact small values and closed-form expressions for ordered and cyclic Ramsey numbers across multiple graph families, extending classical Ramsey theory with order and cyclic constraints. The pattern-based closed forms and the group-theoretic unification via permutational numbers provide structural insight, while the RL lower-bound experiments illustrate an alternative computational technique. These contributions could support further exact determinations and theoretical analysis in structured Ramsey problems.","major_comments":[{"comment":"The sections describing the computational approach and SAT encodings provide no explicit CNF variable mapping or clause list for the order-preserving embedding condition (for R_ord) or the rotational invariance condition (for R_cyc). Without this, or any independent verification such as reproduction of known small classical Ramsey numbers or hand-checked instances for n≤10, the reported tables cannot be confirmed to correctly model the definitions.","section":"Computational sections (around the Kissat experiments)"},{"comment":"The closed-form claims for entire classes (e.g., all R_ord or R_cyc between monotone paths and cycles, or similar pairs) rest on patterns observed in the SAT-derived tables. Because the tables themselves lack cross-checks or encoding validation, these extrapolations are load-bearing but rest on unverified data; a single encoding error would invalidate the pattern-based determinations.","section":"Sections deriving closed forms from computational results"}],"minor_comments":[{"comment":"The abstract mentions the use of Kissat and RLGT but gives no indication of the scale of the computations or the range of graph sizes considered; adding a brief summary of the largest instances solved would improve clarity.","section":null},{"comment":"Notation for the new permutational Ramsey numbers is introduced late; an early diagram or table comparing the three variants (standard, ordered, cyclic) would help readers track the unification argument.","section":null}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the combinatorial scope but the central results are entirely computational; a revision should be required to deposit the exact CNF encodings or solver logs as supplementary material to allow independent checking."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for identifying areas where additional transparency would strengthen the presentation of our computational results. We address the two major comments point by point below.","responses":[{"response":"We agree that the current manuscript lacks sufficient detail on the SAT encoding. In the revised version we will add an explicit description of the variable mapping (one Boolean variable per potential edge in each color together with auxiliary variables for vertex mappings) and the principal clauses that enforce order preservation for R_ord and rotational invariance for R_cyc. We will also include a short verification subsection that reproduces several well-known small classical Ramsey numbers (e.g., R(3,3)=6, R(3,4)=9) using the same encoding framework, and we will supply hand-checkable instances for selected n≤10 cases that match the definitions directly.","revision_made":"yes","referee_comment":"[Computational sections (around the Kissat experiments)] The sections describing the computational approach and SAT encodings provide no explicit CNF variable mapping or clause list for the order-preserving embedding condition (for R_ord) or the rotational invariance condition (for R_cyc). Without this, or any independent verification such as reproduction of known small classical Ramsey numbers or hand-checked instances for n≤10, the reported tables cannot be confirmed to correctly model the definitions."},{"response":"The closed-form statements are indeed pattern-based extrapolations from the computed tables. Once the encoding details and verification results are added (as described in the response to the first comment), the reliability of the underlying data will be established. In the revision we will also state clearly which closed forms are accompanied by combinatorial proofs and which remain conjectural but are supported by exhaustive computation up to a verified bound together with matching theoretical upper and lower bounds derived in the paper. This makes the evidential basis explicit rather than load-bearing on unverified tables.","revision_made":"partial","referee_comment":"[Sections deriving closed forms from computational results] The closed-form claims for entire classes (e.g., all R_ord or R_cyc between monotone paths and cycles, or similar pairs) rest on patterns observed in the SAT-derived tables. Because the tables themselves lack cross-checks or encoding validation, these extrapolations are load-bearing but rest on unverified data; a single encoding error would invalidate the pattern-based determinations."}],"tokens_in":1721,"tokens_out":511,"duration_ms":33427,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper computes new small exact values for ordered Ramsey numbers on monotone paths, cycles, stars, complete graphs and nested matchings with the Kissat solver. It defines cyclic Ramsey numbers as a rotational relaxation, computes more of those, and introduces permutational Ramsey numbers to frame the standard, ordered and cyclic versions inside a single group action. From the tables they extract closed forms for several pairs of classes and give bounds when one graph is connected and the other is a path or cycle. They also run a quick reinforcement-learning experiment for lower bounds.","headline":"New SAT-computed values for ordered and cyclic Ramsey numbers plus some pattern-based closed forms, but the encodings have no visible cross-checks.","tokens_in":2614,"tokens_out":183,"would_cite":false,"duration_ms":27170,"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":"Computational patterns in small ordered and cyclic Ramsey numbers yield exact values for entire classes of graphs such as monotone paths and cycles.","keywords":["ordered ramsey numbers","cyclic ramsey numbers","monotone paths","monotone cycles","ramsey theory","graph colorings"],"falsifier":"An explicit counterexample computation for a pair of larger monotone paths whose ordered Ramsey number differs from the formula suggested by the pattern in smaller cases.","tokens_in":2846,"feed_emoji":"🔢","tokens_out":630,"duration_ms":39087,"temperature":0.7,"pith_summary":"The paper computes small two-color ordered Ramsey numbers for classes including monotone paths, monotone cycles, alternating paths, stars, complete graphs, and nested matchings. It introduces cyclic Ramsey numbers as a relaxation of the ordered version. Structural patterns observed in the results allow exact determinations of all such numbers for several pairs of these classes. Bounds are obtained when one graph is connected and the other is a monotone path or cycle. A unification of the standard, ordered, and cyclic versions is proposed through permutational Ramsey numbers.","feed_headline":"Patterns in small cases fix ordered Ramsey numbers for graph classes","feed_subtitle":"Exact values emerge for all monotone paths and cycles, with a cyclic relaxation and a group-theoretic unification of the variants.","key_machinery":"SAT encodings of the conditions for avoiding monochromatic ordered or cyclically ordered subgraphs, followed by manual detection of patterns across the resulting tables to obtain closed forms for infinite families.","core_discovery":"By computing small instances and identifying recurring structural patterns, all ordered or cyclic Ramsey numbers are determined for several pairs of graph classes, such as monotone paths against monotone cycles. Additional bounds are derived for cases where one argument is a connected graph and the other is a monotone path or cycle. Permutational Ramsey numbers are introduced to place the ordered, cyclic, and classical variants inside a single group-theoretic framework.","pith_inferences":["The same pattern-spotting method from small computations could be tried on other families of extremal numbers to generate conjectured formulas.","The group-theoretic unification opens the possibility of algebraic proofs that replace the computational evidence for the observed formulas.","Because ordered versions require stricter conditions than classical Ramsey numbers, the exact values here supply new lower bounds on classical numbers for the same graphs."],"forward_implications":["Exact closed-form expressions are now available for the ordered Ramsey numbers between any two monotone paths.","The same closed forms hold for the corresponding cyclic Ramsey numbers in those classes.","Upper and lower bounds apply to all ordered and cyclic Ramsey numbers mixing an arbitrary connected graph with a monotone path or cycle.","The permutational formulation places classical, ordered, and cyclic Ramsey numbers inside one common setting."],"fun_headline_variants":["Small cases fix ordered Ramsey numbers for graph class pairs","Patterns fix exact ordered and cyclic Ramsey numbers","SAT solver finds new small two-color ordered Ramsey numbers","Permutational Ramsey numbers unify the variants in group theory"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The structural patterns visible in the computed small instances continue to hold for all larger graphs within the same classes.","fun_headline_variants_meta":{"raw":{"variants":["Small cases fix ordered Ramsey numbers for graph class pairs","Patterns fix exact ordered and cyclic Ramsey numbers","SAT solver finds new small two-color ordered Ramsey numbers","Permutational Ramsey numbers unify the variants in group theory"]},"model":"grok-4.3","cost_usd":0.011917,"raw_usage":{"total_tokens":5307,"prompt_tokens":865,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":119174500,"prompt_tokens_details":{"text_tokens":865,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4382,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":865,"tokens_out":60,"duration_ms":33230,"temperature":1.0,"reasoning_tokens":4382,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-10T07:55:14.154992+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit counterexample computation for a pair of larger monotone paths whose ordered Ramsey number differs from the formula suggested by the pattern in smaller cases.","supporting_citations":[],"review_version":1}