{"id":"c292eecd-01a5-433d-8dff-4cf8843107c8","arxiv_id":"2607.06817","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"SAT-based computation yields exact small reflective and dihedral Ramsey numbers for several ordered graph families, plus closed formulas and conjectures linking them to ordered and cyclic variants.","lead":"The paper computes exact values and lower bounds for small reflective and dihedral Ramsey numbers of paths, cycles, stars, complete graphs and nested matchings via SAT encodings solved by Kissat. It also proves several closed formulas and states conjectures that dihedral numbers often match cyclic ones.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the SAT-timeout premise as the main practical fragility, yet that premise only affects the larger table entries and the open conjectures. The load-bearing mathematical statements are the closed formulas, which are proved by standard combinatorial arguments independent of any solver. Because those proofs check out under ordinary scrutiny and the computational artifacts are public, the ACCEPT verdict stands; no adjustment is warranted.","tokens_in":22825,"tokens_out":305,"duration_ms":4110,"concrete_test":"Independently re-derive the upper-bound induction of Theorem 4.1 for a=3,b=4 (n=7) by hand-enumerating the monochromatic embeddings; if the claimed equality fails for this tiny case, the general formula collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper’s central closed-form claims (Theorem 4.1 and Corollaries 4.2–4.3, 4.16, Proposition 4.12) rest on elementary inductive/pigeonhole arguments that appear correct and do not depend on the SAT pipeline. The computational tables and dihedral=cyclic conjectures are presented as experimental findings with released Kissat certificates and graph6 colorings; the reader’s tooling concern is real but secondary, because the strongest claims are the proved equalities rather than any particular table entry. No internal inconsistency or hidden assumption that would overturn those equalities is visible.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces reflective and dihedral Ramsey numbers as special cases of permutational Ramsey numbers (groups generated by reflection, or by cyclic shift plus reflection). It proves closed formulas for several families, most notably Theorem 4.1: for any connected H of order a and any group Γ on V(H), R(H^Γ,(P_mon_b)^Λ)=1+(a-1)(b-1) when Λ is trivial; Corollaries 4.2–4.3, Propositions 4.10 and 4.12, and Corollaries 4.14–4.16 then give exact reflective (and often dihedral) values for alternating paths or start-central stars versus monotone paths, cycles and complete graphs. The bulk of the work is a SAT encoding (clauses (1)–(2) in §3) solved by Kissat that produces exact small values and lower bounds for the remaining pairs among alternating/monotone paths, monotone cycles, start-central stars, completes and nested matchings (Tables 3–14), together with several conjectures that R_dih coincides with R_cyc for alternating-path arguments.","tokens_in":22929,"tokens_out":745,"duration_ms":7215,"significance":"The work cleanly fills the two natural intermediate cases between ordered and cyclic Ramsey numbers left open by the authors’ earlier framework. Theorem 4.1 and its corollaries are elementary but useful closed formulas that unify several previously scattered observations; the extensive, fully reproducible SAT tables (source code, Kissat logs and graph6 colorings released) supply concrete data that both confirm known ordered/cyclic values and generate plausible dihedral=cyclic conjectures. The contribution is solid computational combinatorics with a modest theoretical advance, appropriate for a specialized discrete-mathematics journal.","major_comments":[],"minor_comments":[{"comment":"Table 2 lists many entries as “—” (no formula known or conjectured). A short remark in §4 explaining why those families resist a simple closed form would help the reader assess the scope of the conjectures that are offered.","section":null},{"comment":"The time limits (2 min generation / 3 min Kissat) and order threshold 30 are stated only in the experimental paragraph of §4. Moving them into §3 (Methodology) would make the computational claims self-contained.","section":null},{"comment":"A few tables (e.g., Table 3, a=3 row) mix exact integers with “≥30”-style lower bounds; a uniform typographic convention for lower bounds would improve readability.","section":null},{"comment":"The heavy dependence on the authors’ prior ordered/cyclic paper [4] and code base [5] is legitimate, but a one-sentence pointer in the introduction to the precise differences in the SAT encoding would clarify novelty for readers unfamiliar with that work.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a natural and competent sequel to the authors’ earlier ordered/cyclic paper. The theoretical core is elementary and correct; the computational contribution is reproducible and useful. I see no reason to demand further theoretical depth for acceptance in a computational-combinatorics venue."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a clean, workmanlike extension of the authors’ recent permutational-Ramsey paper. The genuinely new pieces are the reflective and dihedral numbers themselves (already defined in [4] but not previously tabulated), Theorem 4.1 (any connected H versus a monotone path with trivial group equals the classical 1+(a-1)(b-1)), and the short list of corollaries that immediately give exact reflective values for alternating paths and start-central stars versus monotone paths, cycles and completes. Those proofs are elementary induction-plus-pigeonhole and look correct; they do not lean on the SAT pipeline.\n\nWhat the paper does well is the engineering. They ship a documented Kissat pipeline, graph6 colorings for every lower bound, and the full source. The tables for small alternating-path and star instances are therefore reproducible and will be useful as benchmarks. The dihedral-equals-cyclic conjectures are pattern-driven and left open, which is honest.\n\nSoft spots are minor and proportional. The SAT encoding is the standard Poljak-style one; the only real fragility is the 2 min / 3 min timeout wall and the order-30 cutoff, so a few “≥” entries might hide exact values just beyond the limit. That does not touch the proved equalities. Self-citation of [4] and the earlier code base is heavy but legitimate prior art. Significance stays inside the ordered/cyclic-Ramsey niche; nothing here cracks a classical open problem.\n\nAnyone already working on ordered or cyclic Ramsey numbers will want the tables and the closed formulas. A serious editor should send it to referees; the math is sound, the artifacts are public, and the contribution is real if modest. I would cite the closed formulas and the tables if I needed those numbers.","headline":"Solid computational extension of the authors’ own permutational framework: a few clean closed formulas plus public SAT tables and dihedral=cyclic conjectures; niche but usable.","tokens_in":23593,"tokens_out":455,"would_cite":true,"duration_ms":4783,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D10","05C55"],"pacs":[],"model":"grok-4.5","headline":"Reflective and dihedral Ramsey numbers for paths, stars, cycles and matchings admit closed formulas and small exact values via SAT.","keywords":["Ramsey numbers","reflective Ramsey numbers","dihedral Ramsey numbers","permutational Ramsey numbers","ordered graphs","SAT encoding","alternating paths","start-central stars"],"falsifier":"Exhibit a concrete 2-edge-coloring of Kn that avoids every reflective (respectively dihedral) embedding of the two claimed graphs for any n equal to a reported exact value, or produce a counter-example pair of alternating-path orders that separates the dihedral number from the cyclic number.","tokens_in":23681,"feed_emoji":"⬡","tokens_out":1053,"duration_ms":17310,"temperature":0.7,"pith_summary":"The paper defines reflective and dihedral Ramsey numbers as special cases of permutational Ramsey numbers, in which the allowed embeddings of a graph may reverse its linear order or rotate and reverse a cyclic order. It shows that these numbers often collapse to already-known ordered or cyclic Ramsey numbers when one argument has reflection symmetry, and it proves exact product-type formulas for any connected graph against a monotone path and for start-central stars against monotone paths, cycles and complete graphs. Using a Boolean-SAT encoding of forbidden monochromatic embeddings and the Kissat solver, the authors compute tables of exact values and lower bounds for alternating paths, start-central stars and nested matchings of small order. The computations support several clean conjectures that dihedral numbers coincide with cyclic numbers whenever one argument is an alternating path. The results give a concrete computational and theoretical bridge between ordered, cyclic and classical Ramsey theory for the same families of graphs.","feed_headline":"SAT pins down small reflective and dihedral Ramsey numbers","feed_subtitle":"Closed formulas for paths and stars, plus tables and conjectures that dihedral often equals cyclic.","key_machinery":"Γ-embeddability: a graph H is Γ-embeddable in G when some group element of Γ can be composed with an order-preserving injection to produce a homomorphism into G. The non-existence of monochromatic Γ-embeddings is encoded as a CNF whose clauses forbid every possible increasing image of every group translate of each forbidden edge set; satisfiability of that CNF yields a lower bound and unsatisfiability an upper bound.","core_discovery":"For any connected graph H of order a and any permutation group on its vertices, the permutational Ramsey number against a monotone path of order b (with the trivial group) equals 1+(a-1)(b-1); the same closed formula holds for reflective Ramsey numbers of alternating paths and of start-central stars against monotone paths, and for reflective (hence also ordered, cyclic and dihedral) numbers of start-central stars against monotone cycles and complete graphs. Extensive SAT computations supply exact small reflective and dihedral values for the remaining combinations and motivate the conjecture that the dihedral number of an alternating path against any of the listed families equals the correspo","pith_inferences":["Tailoring the permutation group to the automorphism group of a graph (for example fixing the apex of a fan) may produce still smaller intermediate Ramsey numbers that interpolate between ordered and classical values.","The observed near-equality of reflective and ordered numbers suggests that allowing a single reflection rarely reduces the Ramsey number by more than one for the families studied.","Hardness spikes for odd-order alternating paths against start-central stars under the dihedral group may indicate a combinatorial phase transition worth a separate theoretical analysis."],"forward_implications":["Any connected graph of order a forces a monochromatic monotone path of length b in every 2-edge-coloring of the complete graph on 1+(a-1)(b-1) vertices once the path is required only to be increasing.","Reflective Ramsey numbers of start-central stars against monotone cycles or complete graphs are identical to the corresponding ordered, cyclic and classical numbers, all equal to 1+(a-1)(b-1).","If the dihedral-versus-cyclic conjectures hold, every previously computed cyclic Ramsey number involving an alternating path immediately supplies the matching dihedral number.","The same SAT pipeline yields systematic lower and upper bounds for any other pair of graphs once their reflection or dihedral groups are substituted into the clause generator."],"fun_headline_variants":["SAT solves exact small reflective and dihedral Ramsey numbers","Kissat pins reflective Ramsey values for paths stars cycles","Closed forms plus SAT bounds on dihedral Ramsey numbers","Reflective Ramsey numbers of matchings paths computed via SAT","Dihedral equals cyclic often: SAT tables and path formulas"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The SAT encoding together with Kissat’s unsatisfiability answers correctly certify that no avoiding 2-edge-coloring exists, and the chosen time limits do not turn an exact value into a mere lower bound.","fun_headline_variants_meta":{"raw":{"variants":["SAT solves exact small reflective and dihedral Ramsey numbers","Kissat pins reflective Ramsey values for paths stars cycles","Closed forms plus SAT bounds on dihedral Ramsey numbers","Reflective Ramsey numbers of matchings paths computed via SAT","Dihedral equals cyclic often: SAT tables and path formulas"]},"model":"grok-4.5","effort":"low","cost_usd":0.006816,"raw_usage":{"total_tokens":1866,"prompt_tokens":1047,"num_sources_used":0,"completion_tokens":80,"cost_in_usd_ticks":68160000,"prompt_tokens_details":{"text_tokens":1047,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":739,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1047,"tokens_out":80,"duration_ms":6590,"temperature":1.0,"reasoning_tokens":739,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T15:55:26.781005+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a concrete 2-edge-coloring of Kn that avoids every reflective (respectively dihedral) embedding of the two claimed graphs for any n equal to a reported exact value, or produce a counter-example pair of alternating-path orders that separates the dihedral number from the cyclic number.","supporting_citations":[],"review_version":2}