{"id":"5e32b814-4b4d-47a6-b846-4821d553aeb4","arxiv_id":"1908.04129","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The anti-Ramsey numbers of star forests and double stars are determined exactly, and the anti-Ramsey number of any linear forest is determined up to an additive constant.","lead":"This paper calculates the maximum number of colors you can use to color the edges of a complete graph without forcing a rainbow subgraph that is a star forest, a linear forest, two disjoint 4-vertex paths, or a double star. It settles exact answers for star forests and double stars, and gives a close approximation for all linear forests.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1 is internally inconsistent with Theorem 5(6): for F=P2∪P3 it gives a lower bound of 4 while the cited Theorem 5(6) gives 2.","rationale":"The Reader's conditional verdict is reasonable, but the most load-bearing issue I find is not the omitted s<6 case; it is a false supporting proposition in the same section. The paper's own Theorem 5(6) contradicts Proposition 1 for F=P2∪P3. This matters because Proposition 1 is the stated lower bound for all linear forests, and Corollary 1, Theorem 12, and Theorem 9 all build on it. The star-forest Theorem 8 proof appears sound; in particular, the extension step in Case 1 works because after recoloring all C(v0) colors to a single color, a rainbow F−K1,p1 under c′ contains at most one edge whose original color lies in C(v0), making the counting of available colors at v0 sufficient. The s<6 omission in Theorem 12 is real, but it is an acknowledged missing subcase in an upper-bound argument; the false lower-bound proposition is a definite error that should be corrected, for instance by excluding the first construction when F contains a P2 component or by supplying a valid proof. The double-star and 2P4 arguments did not reveal an independent issue. I therefore keep the same overall recommendation: conditional acceptance pending revision.","tokens_in":16346,"tokens_out":59193,"duration_ms":615457,"concrete_test":"Fix F=P2∪P3. (1) Verify Theorem 5(6) for t=1 by an independent argument or exact search: determine ar(Kn,P2∪P3) for n≥5. If it is 2, Proposition 1's lower bound of 4 is false; if it is at least 4, then the quoted Theorem 5(6) is wrong. (2) In the first construction of Proposition 1, explicitly exhibit the rainbow F: take the two disjoint vertices outside K3 and the edge between them (colored with the extra color), together with any two edges of the rainbow K3 sharing a vertex. This isolates the exact false step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3's Proposition 1 is false as stated, and this is internally inconsistent with the paper's own Theorem 5(6). For F=P2∪P3 (p1=2,p2=3), Proposition 1's first lower-bound term is C(5−2,2)+1=4. Yet Theorem 5(6) with t=1 says ar(Kn,P2∪P3)=2. The flaw is in the first construction: color a K_{Σp_i−2}=K3 rainbow and all other edges with one new color. This coloring contains a rainbow P2∪P3: take a P3 on two edges of the rainbow K3 and the disjoint extra-colored edge between the two outside vertices. Hence the claimed lower bound is not established. Since Proposition 1 is the stated lower bound for the linear-forest section and is invoked by Corollary 1 and Theorem 12, the proof chain for Theorem 9 is unsound as written, even if the asymptotic statement may be repairable by deleting or correcting the first term.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies anti-Ramsey numbers ar(K_n,F) for forests F. It claims an exact formula for star forests (Theorem 8), an approximate formula for linear forests (Theorem 9), an exact value for 2P4 (Theorem 10), and an exact formula for double stars (Theorem 11). The proofs use prior results of Jiang, Simonovits-Sós, Gilboa-Roditty, and Lidický-Liu-Palmer, together with counting arguments on representing subgraphs, rainbow paths, and degree-constrained colorings.","tokens_in":10,"tokens_out":18224,"duration_ms":257313,"significance":"If correct, the star-forest theorem would be a genuine extension of known matching and star results, and the double-star theorem would provide new exact values for a natural family of trees. The paper also gives a plausible route to the asymptotic anti-Ramsey number of all linear forests. Strengths include explicit lower-bound colorings and a detailed induction for star forests. However, the linear-forest section contains a false lower-bound proposition and an explicitly omitted small case, and Section 4 relies on an unjustified reduction to triangles, so the paper is not yet in publishable form.","major_comments":[{"comment":"The first lower-bound construction is invalid. For F=P2∪P3, the construction colors a K_{Σp_i−2}=K3 rainbow and gives one new color to all remaining edges. This coloring contains a rainbow P2∪P3: choose a P3 on two edges of the rainbow K3 and, since n≥5, the disjoint edge between the two vertices outside the K3. Hence the claimed lower bound of 4 is not established, and it contradicts Theorem 5(6), which gives ar(K_n,P2∪P3)=2 for large n. Because Corollary 1 and Theorem 12 use Proposition 1 for their lower bounds, the lower-bound side of the linear-forest results is unsound as written. The asymptotic coefficient in Theorem 9 may be recoverable by deleting or correcting the first term, but the proposition as stated is false.","section":"Section 3, Proposition 1"},{"comment":"The proof is carried out only for s≥6; the text says the case s<6 'can be proved by the similar arguments but need to distinguish more cases as in [20]' without giving any details. Since Theorem 12 is stated for all s≥1 and Theorem 9 depends on it, the missing verification of s=1,…,5 is load-bearing. The authors should either supply a complete proof for those cases or explicitly restrict the theorem and explain how Theorem 9 is obtained.","section":"Section 3, Theorem 12"},{"comment":"After applying Theorem 3, the proof states 'Then k=3' without justification. A graph in Ω2 can consist of two disjoint cycles of length at least 4 (for example C4∪C4), which contains no triangle plus a disjoint cycle; hence the reduction to a rainbow C3∪C_l is not immediate from Theorem 3. The subsequent case analysis depends entirely on this reduction. Please provide a proof of the reduction or a precise citation of a strengthening of Theorem 3 that guarantees a rainbow triangle together with a vertex-disjoint cycle.","section":"Section 4, Lemma 1 and Theorem 10"}],"minor_comments":[{"comment":"In the statement, p_{t−1} is undefined when t=1; the formula for the second term should be restricted to t≥2, since the proof already treats t=1 separately.","section":"Section 2, Theorem 8"},{"comment":"The notation K_s + K_{n-s} (and similarly K_{i-1}+K_{n-i+1}) is ambiguous: it suggests the join of two complete graphs, but the subsequent instruction to color the edges of K_{n-s} with only a few colors indicates a different intended coloring. Please clarify the notation.","section":"Sections 2 and 3, lower-bound constructions"},{"comment":"The displayed cycle C_3^4 = zx1x3x1z contains a repeated vertex; it should presumably be zx1x3z.","section":"Section 4, Lemma 1, Case 2"},{"comment":"The displayed formulas in Theorem 5 contain apparent typographical errors, such as 't/2' where a binomial coefficient is likely intended; please proofread the quoted statements against the original sources.","section":"Theorem 5, quoted formulas"},{"comment":"Fact 1 is described as trivial but is not immediate to me: a rainbow K6 on six vertices alone cannot host two vertex-disjoint P4s, so the fact relies on the colors of edges incident to the remaining vertices. A short justification would help.","section":"Section 4, Fact 1"}],"recommendation":"major_revision","confidential_remarks":"The false Proposition 1 appears repairable by removing or correcting the first lower-bound term; the omitted s<6 cases in Theorem 12 are a completeness issue; and the k=3 step in Section 4 may follow from a known strengthening of Jin-Li's theorem but needs to be stated explicitly. These are nontrivial but fixable problems, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a mixed bag: the exact results for star forests (Thm 8), 2P4 (Thm 10), and double stars (Thm 11) are real contributions, and the proofs check out in the main counting steps. The authors adapt known techniques cleanly, and they credit Jiang, Simonovits–Sós, Gilboa–Roditty, and Lidický–Liu–Palmer properly. If I worked in anti-Ramsey theory, I would want these theorems in the literature.\n\nThe soft spot is Section 3. Proposition 1, the lower bound for linear forests, is false as stated. For F = P2∪P3 (p1=2, p2=3), the first term gives (5−2 choose 2)+1 = 4, while the paper's own Theorem 5(6) says ar(Kn, P2∪P3) = 2. The construction fails: color a K3 rainbow and all other edges with one extra color; that coloring contains a rainbow P2∪P3 (two adjacent rainbow edges plus a disjoint extra-colored edge). The same issue arises for 2P2. So the statement needs repair, and the proof of the first lower bound is invalid as written.\n\nWhat does this do to the main results? Corollary 1 and the lower bound of Theorem 12 both invoke Proposition 1. However, the lower bound actually needed for the linear forest asymptotics is the second term of Proposition 1, coming from the join construction, and that construction appears sound. So I expect Theorem 9 to survive if the authors drop or fix the first term and re-route the proof through the second construction. But it is not something a referee should have to reverse-engineer.\n\nThere is also the acknowledged gap in Theorem 12: the proof only handles s ≥ 6 and punts on s < 6 with \"similar arguments.\" Since Theorem 9 depends on Theorem 12, that needs to be filled in the revision. It may be routine, but it is load-bearing.\n\nThe rest of the paper looks solid. The 2P4 proof is a case analysis but coherent; the double-star argument via the representing subgraph and the lemma on degree-color constraints is neat.\n\nBottom line: worth sending to a serious referee. The revision should fix Proposition 1 and complete the small-s cases. I would accept it for peer review.","headline":"The star-forest, 2P4, and double-star results are solid, but the linear-forest section contains a false lower-bound proposition (P2∪P3 gives 4 while Theorem 5(6) gives 2); the asymptotics likely survive a fix.","tokens_in":17090,"tokens_out":6200,"would_cite":true,"duration_ms":61263,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C55","05C15","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper determines exactly how many colors in a complete graph force a rainbow copy of any star forest, and pins down the approximate threshold for linear forests.","keywords":["anti-Ramsey number","rainbow subgraph","star forest","linear forest","double star","edge-coloring","representing subgraph","Turán number"],"falsifier":"A computer search for a 17-coloring of $K_8$ with no rainbow $2P_4$ would directly test Theorem 10; finding one would disprove it. For the star-forest formula, checking a specific small case such as $F=K_{1,3}\\cup K_{1,3}$ at the stated $n$ bound against the formula would expose any error in the induction. For the linear-forest theorem, writing out the omitted $s<6$ cases is the direct check; a counterexample to the claimed $(\\sum_i \\lfloor p_i/2\\rfloor - \\epsilon)n + O(1)$ form in one of those small cases would falsify the theorem.","tokens_in":16128,"feed_emoji":"🌈","tokens_out":9243,"duration_ms":80233,"temperature":0.7,"pith_summary":"An edge-coloring of a complete graph is rainbow-free for a graph $G$ when no copy of $G$ has all its edges in distinct colors; the anti-Ramsey number $ar(K_n,G)$ is the largest number of colors that can be used without creating such a rainbow $G$. This paper proves an exact formula for $ar(K_n,F)$ when $F$ is a forest made of stars, valid for all large $n$, in terms of the star sizes and the number of stars. It also proves that for forests made of paths, the anti-Ramsey number is $(\\sum_i \\lfloor p_i/2 \\rfloor - \\epsilon)n + O(1)$, where $\\epsilon$ depends only on whether all path lengths are odd, pinning the linear term exactly but leaving the constant unspecified. Two other exact results are included: $ar(K_n, 2P_4) = \\max\\{2n-2, 16\\}$ for $n\\ge 8$, and an exact formula for double stars $S_{p,q}$ for large $n$.","feed_headline":"Star forests get exact anti-Ramsey numbers","feed_subtitle":"Linear forests are pinned down up to a constant, and exact thresholds are given for two paths and double stars.","key_machinery":"The main device is the representing subgraph: a spanning subgraph of the colored complete graph that keeps exactly one edge of each color, so counting its edges counts the colors. The upper-bound arguments impose degree constraints on this subgraph: if some vertex is incident to many distinct colors, one can detach a star component and build a rainbow forest by induction; otherwise every vertex has bounded color-degree, and counting edges in the representing subgraph against the forbidden forest forces a rainbow copy. For star forests, the extremal colorings come from coloring a join $K_{i-1} + K_{n-i+1}$ rainbow and coloring the remaining clique with the star anti-Ramsey number. For linear forests, the proof uses a longest-rainbow-path method, partitioning the leftover vertices into three classes and bounding the edges in each via extremal estimates on paths and long cycles. For double stars, a degree lemma bounds the number of colors when every vertex sees at most a prescribed number of distinct colors, extending the star argument.","core_discovery":"The central claim is Theorem 8: for a star forest $F = \\bigcup_{i=1}^t K_{1,p_i}$ with $p_1\\ge 3$ and $p_1\\ge \\cdots \\ge p_t\\ge 1$, whenever $n\\ge 3t^2(p_1+1)^2$, the anti-Ramsey number is the maximum of two kinds of terms: $(i-1)n - \\binom{i}{2} + \\lfloor (p_i-2)(n-i+1)/2 \\rfloor + 1$ over those stars with at least two leaves, and $(t-2)n - \\binom{t-1}{2} + r$, where $r=1$ if $p_{t-1}=1$ and $r=2$ otherwise. The paper further claims that a linear forest with path orders $p_1,\\dots,p_k$ satisfies $ar(K_n,F) = (\\sum_i \\lfloor p_i/2 \\rfloor - \\epsilon)n + O(1)$, with $\\epsilon=1$ when all $p_i$ are odd and $\\epsilon=2$ otherwise. It also establishes the exact values for $2P_4$ and for double stars, and shows that a natural additive conjecture for disjoint tree copies fails for paths of length at least four.","pith_inferences":["Beyond the paper: the star-forest formula mirrors the Tur\\'an number of the same forest but with $p_i-2$ in place of $p_i-1$, suggesting a general template in which the rainbow threshold is the extremal edge bound with the per-component parameter lowered by one and one color added.","Beyond the paper: completing the omitted small cases $s<6$ in the linear-forest proof would convert the $O(1)$ into an exact formula, and those omitted cases appear to be the only obstacle to a fully explicit constant.","Beyond the paper: the same representing-subgraph and degree-lemma arguments could be tested on spiders with three legs, where the paper's closing conjecture predicts a sharp lower bound; checking that conjecture on a concrete spider is a direct next step."],"forward_implications":["For any fixed star forest, the exact anti-Ramsey number is now known for all sufficiently large complete graphs, so the threshold color count can be read directly from the star sizes.","The linear-forest formula shows that the natural additive conjecture for disjoint tree copies fails for paths of length at least 4.","The exact value for $2P_4$ settles the smallest multi-path case not covered by earlier results, giving $2n-2$ once $n\\ge 9$.","Double stars, the simplest non-star trees with two centers, have anti-Ramsey numbers matching the star threshold when one leaf class is smaller, and a slightly different threshold when the two centers have equal leaf counts.","The results provide exact data points connecting anti-Ramsey numbers to Tur\\'an-type extremal numbers for disconnected forests."],"supporting_citations":[{"why":"supplies the anti-Ramsey number for single stars used in the star-forest lower-bound construction and in the double-star degree lemma.","marker":"[12]"},{"why":"gives the complementary star anti-Ramsey result used in Theorem 1's bounds.","marker":"[18]"},{"why":"provides the exact anti-Ramsey number for matchings used as the lower bound when the last star has one leaf.","marker":"[19]"},{"why":"provides the anti-Ramsey numbers for small forests such as $K_{1,2}\\cup K_2$ used in the second lower bound and in the star-forest case analysis.","marker":"[9]"},{"why":"gives the path anti-Ramsey formula and the longest-path method that the linear-forest proof adapts.","marker":"[20]"},{"why":"gives the Tur\\'an numbers of star and linear forests used to compare the anti-Ramsey threshold with the extremal edge count.","marker":"[15]"},{"why":"provides the anti-Ramsey number for two vertex-disjoint cycles, the starting point for the exact $2P_4$ proof.","marker":"[14]"},{"why":"supplies the extremal bounds on paths and long cycles used to bound edges in the linear-forest partitioning argument.","marker":"[5]"}],"fun_headline_variants":["Anti-Ramsey forests: exact stars, approximate paths","Star forest anti-Ramsey numbers now exact","Forest anti-Ramsey: stars exact, paths within a constant","Exact anti-Ramsey values for star and double-star forests","Forest anti-Ramsey: stars nailed, paths pinned to O(1)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the linear-forest upper bound is written only for the case where the single even path has at least twelve vertices; the smaller cases are dismissed as provable by similar arguments without being written out, and the approximate linear-forest formula depends on those omitted cases.","fun_headline_variants_meta":{"raw":{"variants":["Anti-Ramsey forests: exact stars, approximate paths","Star forest anti-Ramsey numbers now exact","Forest anti-Ramsey: stars exact, paths within a constant","Exact anti-Ramsey values for star and double-star forests","Forest anti-Ramsey: stars nailed, paths pinned to O(1)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000831,"raw_usage":{"total_tokens":3642,"prompt_tokens":969,"completion_tokens":2673,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":2586}},"tokens_in":585,"tokens_out":2673,"duration_ms":20710,"temperature":1.0,"reasoning_tokens":2586,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:53:08.291425+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A computer search for a 17-coloring of $K_8$ with no rainbow $2P_4$ would directly test Theorem 10; finding one would disprove it. For the star-forest formula, checking a specific small case such as $F=K_{1,3}\\cup K_{1,3}$ at the stated $n$ bound against the formula would expose any error in the induction. For the linear-forest theorem, writing out the omitted $s<6$ cases is the direct check; a counterexample to the claimed $(\\sum_i \\lfloor p_i/2\\rfloor - \\epsilon)n + O(1)$ form in one of those small cases would falsify the theorem.","supporting_citations":[{"cited_title":"Jiang, Edge-coloring with no large polychromatic st ars, Graphs Combin","cited_arxiv_id":null,"evidence_quote":"supplies the anti-Ramsey number for single stars used in the star-forest lower-bound construction and in the double-star degree lemma."},{"cited_title":"Montellano-Ballesteros, On totally multicolore d stars, J","cited_arxiv_id":null,"evidence_quote":"gives the complementary star anti-Ramsey result used in Theorem 1's bounds."},{"cited_title":"Schiermeyer, Rainbow numbers for matchings and comp lete graphs, Discrete Math","cited_arxiv_id":null,"evidence_quote":"provides the exact anti-Ramsey number for matchings used as the lower bound when the last star has one leaf."},{"cited_title":"Gilboa and Y","cited_arxiv_id":null,"evidence_quote":"provides the anti-Ramsey numbers for small forests such as $K_{1,2}\\cup K_2$ used in the second lower bound and in the star-forest case analysis."},{"cited_title":"Simonovits and V.T","cited_arxiv_id":null,"evidence_quote":"gives the path anti-Ramsey formula and the longest-path method that the linear-forest proof adapts."},{"cited_title":"Lidick´ y, H","cited_arxiv_id":null,"evidence_quote":"gives the Tur\\'an numbers of star and linear forests used to compare the anti-Ramsey threshold with the extremal edge count."},{"cited_title":"Jin and X","cited_arxiv_id":null,"evidence_quote":"provides the anti-Ramsey number for two vertex-disjoint cycles, the starting point for the exact $2P_4$ proof."},{"cited_title":"Erd˝ os and T","cited_arxiv_id":null,"evidence_quote":"supplies the extremal bounds on paths and long cycles used to bound edges in the linear-forest partitioning argument."}],"review_version":1}