{"id":"9b6c7e33-ca6c-4a9f-be97-f1c49ed31455","arxiv_id":"2607.03243","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"In 3-edge-coloured complete graphs, R_3^1(P_n) equals 3n/2 + O(1) and R_3^2(C_n) equals 3n/2 + o(n) for even n.","lead":"The paper defines a new Ramsey number that allows a few colour changes in paths and cycles, then proves that three-coloured complete graphs already force long paths with one change and even cycles with two changes at size roughly 3n/2. This recovers the classical two-colour path Ramsey growth while linking to open monochromatic path-cover conjectures.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the sole non-constructive dependence (regularity-based cycle Ramsey numbers) and notes that it is inherited rather than introduced by the paper. The new combinatorial core—reduction to a split colouring followed by a double-counting argument over monochromatic path covers—is self-contained, size-explicit, and free of additional asymptotic losses. Consequently the stated bounds stand exactly as claimed, and no adjustment to the ACCEPT verdict is warranted.","tokens_in":21297,"tokens_out":397,"duration_ms":3969,"concrete_test":"Verify the arithmetic identity used after Lemma 16 in the proof of Theorem 17: sum of the twelve path lengths is at least 8N+4, which for N=⌈(3n-2)/2⌉ is at least 12n-8; confirm that the same identity continues to force a path of length ≥n when the four part sizes are allowed to differ by O(1) (as occurs when 4 does not divide n).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims of Theorems 8 and 9 rest on a clean reduction (Lemmas 15/19) to a four-partite split colouring whose good-path/cycle existence is then settled by an elementary counting argument that invokes only Pokrovskiy’s three-path cover (Lemma 13) and a bipartite path-extension lemma (Lemma 16). Both external ingredients are fully constructive and hold for all n; the only non-constructive input is the three-colour cycle Ramsey number (Theorem 2), which is used solely to force the colouring into the split form and is already known to hold for all sufficiently large n. No internal gap, hidden size assumption, or parity obstruction appears in the reduction or the counting step that would invalidate the stated asymptotic or additive-constant bounds.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces the Ramsey-type parameter R_q^k(G), the smallest N such that every q-edge-colouring of K_N contains a copy of G with at most k colour-change vertices. Focusing on three colours, it proves that floor((3n-2)/2) ≤ R_3^1(P_n) ≤ 3⌈n/2⌉ + 2 (and ≤ 3n/2 when 4|n) for all sufficiently large n (Theorem 8), and that R_3^2(C_n) = (3/2 + o(1))n for all sufficiently large even n (Theorem 9). The lower bound is realised by an explicit four-partite colouring whose template is a 2-coloured K_4. The upper bounds proceed by a reduction (Lemmas 15 and 19) that either produces a good path/cycle directly or forces a split four-partite colouring; the latter case is settled by an elementary counting argument that invokes Pokrovskiy’s three-path cover and a bipartite path-extension lemma (Lemmas 13 and 16).","tokens_in":21505,"tokens_out":902,"duration_ms":6411,"significance":"The work cleanly interpolates between classical multicolour Ramsey numbers for paths and cycles and the monochromatic-path-cover conjectures of Gyárfás and Erdős–Gyárfás–Pyber. The asymptotic coincidence of R_3^1(P_n) with the two-colour path Ramsey number is striking and suggests that a single colour change exactly offsets the third colour. All arguments after the invocation of the known large-n three-colour cycle Ramsey numbers are fully combinatorial, constructive, and free of further regularity; the lower-bound construction is elementary and exact. The new parameter R_q^k and the open problems posed in the concluding section (odd cycles, spanning trees, κ_q) open a natural research direction.","major_comments":[],"minor_comments":[{"comment":"The abstract and Theorem 8 state the path upper bound as 3⌈n/2⌉ + 2, while the body (after Lemma 15) notes that N ≥ 3⌈n/2⌉ + 2 already forces the hypotheses of the reduction; a single sentence clarifying the precise additive constant for all residue classes of n would improve readability.","section":"Abstract / Theorem 8"},{"comment":"In the proof of Lemma 15 the case k = ⌈n/2⌉ + 1 allows the two monochromatic cycles to share a single vertex; the subsequent argument that this immediately yields a good path of length > n is correct but slightly terse—one extra sentence would help the reader.","section":"Lemma 15"},{"comment":"Remark 20 correctly isolates the only parity-dependent steps in Lemma 19; it would be useful to restate explicitly that Theorem 21 already holds for odd n, so that the sole remaining obstacle for odd cycles is the bipartite-endpoint argument of Case 1.","section":"Remark 20 / Section 6"},{"comment":"A few typographical inconsistencies appear (e.g., “Theoren 5”, “Op 1q”, missing spaces around some set-builder braces). A light copy-edit pass would remove them.","section":"Throughout"}],"recommendation":"accept","confidential_remarks":"The manuscript is a solid, self-contained contribution that sits comfortably within the scope of a strong combinatorics journal. The dependence on regularity-based cycle Ramsey numbers is inherited and fully acknowledged; it does not affect the novelty or correctness of the new reduction. I see no reason to delay acceptance."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper introduces the natural parameter R_q^k(G) and settles the three-colour case for paths with one colour change and even cycles with two. The punchline is that allowing a single colour change exactly cancels the extra colour: R_3^1(P_n) sits at 3n/2 + O(1), matching the classical two-colour path Ramsey number, and the even-cycle version is (3/2 + o(1))n.\n\nWhat is new is the parameter itself and the reduction to a four-partite split colouring (Lemmas 15 and 19). Once the colouring is forced into that template, an elementary double-counting argument that uses only Pokrovskiy’s three-path cover and a bipartite path-extension lemma extracts the desired good path or cycle. The lower-bound construction is simple and exact. The proofs are fully written out and combinatorial; they do not invent new regularity machinery.\n\nThe soft spots are exactly the ones the authors flag. Upper bounds inherit the “sufficiently large n” threshold and non-constructive character of the three-colour cycle Ramsey numbers (Benevides–Skokan, Kohayakawa–Simonovits–Skokan). Odd cycles are left open, and the o(n) term for even cycles is not improved to O(1). Neither gap undermines the stated theorems. Circularity is zero; the external black boxes are independent published results.\n\nThis is for people who work on multicolour Ramsey numbers, monochromatic path covers, or colour-change variants of classical problems. It is a clean, self-contained contribution that a serious referee should see. I would accept it for peer review and expect only minor polishing.","headline":"Clean three-colour colour-change Ramsey numbers for paths (additive constant) and even cycles (o(n)), with solid combinatorial proofs that inherit only the known large-n cycle Ramsey thresholds.","tokens_in":22090,"tokens_out":457,"would_cite":true,"duration_ms":4882,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C55","05C38"],"pacs":[],"model":"grok-4.5","headline":"In three-edge-coloured complete graphs a path of length n with one colour change is already forced once the graph has only about 3n/2 vertices.","keywords":["Ramsey numbers","colour changes","paths","cycles","three-edge-colourings","monochromatic path covers","split colourings"],"falsifier":"An explicit three-edge-colouring of a complete graph on roughly 3n/2 + ω(1) vertices that contains neither a path of length n with one colour change nor (for even n) a cycle of length n with two colour changes would refute the claimed asymptotic thresholds.","tokens_in":22227,"feed_emoji":"🔀","tokens_out":1064,"duration_ms":8002,"temperature":0.7,"pith_summary":"Classical Ramsey theory asks for monochromatic copies of a graph. This paper relaxes that demand for paths and cycles: it allows a handful of colour changes and measures how many vertices are then needed to force the structure. The new parameter R_q^k(G) is the smallest N such that every q-edge-colouring of K_N contains a copy of G with at most k colour-change vertices. For three colours the authors prove that a path on n vertices with at most one colour change appears already at N = 3n/2 + O(1). For even cycles the same asymptotic threshold works when two colour changes are allowed. The results sit between the classical three-colour Ramsey numbers (which grow like 2n) and the two-colour path Ramsey number (which is already 3n/2), showing that one extra colour change exactly cancels the cost of the third colour. The proofs combine known large-n Ramsey numbers for monochromatic cycles with elementary splitting arguments and a bipartite matching lemma that produces long paths or cycles once a colouring has a rigid four-partite structure.","feed_headline":"One colour change drops three-colour path Ramsey to 3n/2","feed_subtitle":"Allowing a single switch cancels the third colour; even cycles need only two switches","key_machinery":"The auxiliary parameter R_q^k(G) together with a two-step reduction: first a Ramsey-plus-splitting lemma that either produces the desired path/cycle or forces a rigid four-partite colouring of the complete graph; second an elementary counting argument (via Pokrovskiy’s three-path cover) that extracts a short colour-change path or cycle from every such four-partite colouring.","core_discovery":"For every sufficiently large n one has floor((3n-2)/2) ≤ R_3^1(P_n) ≤ 3⌈n/2⌉ + 2 (and ≤ 3n/2 when 4 divides n). For every sufficiently large even n one has R_3^2(C_n) = (3/2 + o(1))n. In other words, allowing a single colour change for paths, or two for even cycles, reduces the three-colour Ramsey threshold from roughly 2n down to the classical two-colour threshold 3n/2.","pith_inferences":["If the odd-cycle case can be settled at the same 3n/2 threshold, colour-change Ramsey numbers for cycles would behave uniformly, in contrast to ordinary Ramsey numbers where odd cycles require roughly twice as many vertices.","The same four-partite template that supplies the lower bound may be the only obstruction; a refined analysis of edges inside the parts could replace the o(n) error by an absolute constant for even cycles.","The definition of κ_q(G) (the minimal number of colour changes needed for a spanning copy) opens a new extremal question for trees, planar graphs and regular graphs that the paper only sketches."],"forward_implications":["Allowing one colour change exactly offsets the penalty of a third colour for paths, recovering the classical two-colour growth rate.","The same asymptotic holds for even cycles with two colour changes, suggesting a broader pattern for sparse graphs.","A Hamilton path with at most two colour changes in any three-edge-coloured K_n would imply a cover by three monochromatic paths, recovering a known theorem of Pokrovskiy.","The four-partite split colourings that arise as extremal candidates become the natural objects for studying the remaining odd-cycle case."],"fun_headline_variants":["One colour switch drops 3-colour path Ramsey to 3n/2","Single change cuts three-colour paths to the 3n/2 threshold","Paths with one switch match two-colour Ramsey size 3n/2","Two colour changes let even cycles hit (3/2+o(1))n in three colours","Allowing one change lowers R_3(P_n) from ~2n to 3n/2"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The upper-bound arguments rely on the known three-colour Ramsey numbers for cycles, which themselves hold only for sufficiently large n and rest on the regularity lemma.","fun_headline_variants_meta":{"raw":{"variants":["One colour switch drops 3-colour path Ramsey to 3n/2","Single change cuts three-colour paths to the 3n/2 threshold","Paths with one switch match two-colour Ramsey size 3n/2","Two colour changes let even cycles hit (3/2+o(1))n in three colours","Allowing one change lowers R_3(P_n) from ~2n to 3n/2"]},"model":"grok-4.5","effort":"low","cost_usd":0.005248,"raw_usage":{"total_tokens":1502,"prompt_tokens":845,"num_sources_used":0,"completion_tokens":117,"cost_in_usd_ticks":52480000,"prompt_tokens_details":{"text_tokens":845,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":540,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":845,"tokens_out":117,"duration_ms":5167,"temperature":1.0,"reasoning_tokens":540,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T03:51:11.410756+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit three-edge-colouring of a complete graph on roughly 3n/2 + ω(1) vertices that contains neither a path of length n with one colour change nor (for even n) a cycle of length n with two colour changes would refute the claimed asymptotic thresholds.","supporting_citations":[],"review_version":1}