{"id":"e3b9b91e-2beb-4d7f-9f2d-349e53156762","arxiv_id":"2505.14386","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper determines extremal edge counts (minimum, sum, product) for rainbow graph collections avoiding both a fixed graph F and a matching of size s+1.","lead":"This paper studies how many edges a collection of graphs can have while containing no rainbow copy of a fixed graph F and no rainbow matching of size s+1. It determines the exact or asymptotic values for three density measures, covering all possible choices of F.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2(ii) proof for s≥5 gives a t-dependent bound, not the stated s-dependent bound; the missing step is showing extra colors are empty.","rationale":"The reader identified Lemma 2.3(ii) as the weakest assumption. That concern is real but appears fixable: Lemma 2.3(ii) is likely true via a different counting argument, since a non-strong color with s(n−s) edges forces at least s vertices of degree at least n/(2s). The more concrete and internally inconsistent issue is in the proof of Theorem 1.2(ii) for s≥5, where the final displayed bound is t⌊n^2/4⌋ rather than the s⌊n^2/4⌋ claimed by the theorem. The proof never shows that colors beyond the first s contribute negligibly. Because Theorem 1.2(ii) is one of the paper's main exact results, this gap must be closed before the result can be accepted as proven. The overall verdict should remain conditional: the paper is plausible and mostly carefully written, but this specific proof step is unsupported as written.","tokens_in":16400,"tokens_out":30360,"duration_ms":300106,"concrete_test":"For s=5 and t=6, prove or disprove that any nested rainbow {K3,M6}-free collection (G1⊇...⊇G6) has |E(G6)|=O(n). One direct check: if G6 contains an edge e, show that G1,...,G5 contain a rainbow matching of size 5 disjoint from e; this would force a rainbow M6. If the derivation succeeds, the claimed bound follows and the proof needs only a correction. If it fails, construct a collection with Σ_{i=1}^6 |E(Gi)|>5⌊n^2/4⌋, which would refute Theorem 1.2(ii) as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.2(ii), case s≥5, the authors use the KSSV bound for the first four colors to obtain |E(G1)|+...+|E(G4)|≤4⌊n^2/4⌋, hence |E(Gi)|≤⌊n^2/4⌋ for every i≥4. They then conclude the desired bound is Σ_{i=1}^t |E(Gi)|≤t⌊n^2/4⌋. But the theorem states ex^Σ_t(n,{K3,M_{s+1}})=ex^Σ_s(n,K3), and the introduction explicitly records ex^Σ_s(n,K3)=s⌊n^2/4⌋ for s≥4. For t>s, the bound t⌊n^2/4⌋ exceeds the claimed value by (t−s)⌊n^2/4⌋. The argument never uses s to control colors beyond the fourth, so it does not prove that G_{s+1},...,G_t are empty or sparse. This is not a stylistic gap: the displayed conclusion of the proof contradicts the statement of the theorem when t>s. The missing load-bearing assertion is that every edge in any color beyond the first s is forbidden by the rainbow M_{s+1}-free condition, or at least that the total contribution of those colors is O(n). The cases s=3,4 do handle the extra colors, but the s≥5 case does not.","agreement_with_reader":"partial"},"referee_report":null,"author_rebuttal":null,"desk_editor":null,"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-07T15:38:00.104325+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}