{"id":"607ba15c-54cb-4db8-b6d8-1f985f586614","arxiv_id":"2608.04845","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every fixed non-matching graph H and fixed colour number ℓ, the exponential counting rate of rainbow-H-free ℓ-colourings of G(n,p) is determined on both sides of the threshold p=n^{-1/m2(H)} by a deterministic template entropy.","lead":"For any fixed graph H, this paper determines how many ways there are to colour the edges of a random graph using a fixed palette while avoiding a rainbow copy of H. It proves a sharp threshold: below it almost all edges can be coloured freely, above it the count is governed by a deterministic optimisation problem.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the argument is internally consistent and the external KŁR input is applied in a standard regime.","rationale":"The reader's verdict of ACCEPT with high confidence seems justified. The paper's central claim is proven by a complete chain: a deterministic entropy optimisation, a sparse random-graph lower bound, and a dense transference via sparse regularity and KŁR. I checked the places where such arguments usually break: the Hall-palette inequality (Lemma 2.4 and Proposition 2.5), the robust entropy dichotomy (Proposition 3.2), the certificate counting in §4.2, and the variance estimate in Lemma 4.1. All are internally consistent. The reader's weakest_assumption correctly identifies Lemma 2.2 as the main external input; I agree that this is the most load-bearing external citation, but it is applied in a standard way and I do not see a concrete failure. Hence no adjustment to the verdict is needed.","tokens_in":16126,"tokens_out":43563,"duration_ms":416810,"concrete_test":"Verify Lemma 2.2 against the CGSS statement of the KŁR theorem: confirm that it applies for all p ≥ A n^{-1/m2(H)} with disjoint sets of size at least n/(2M), and not only for p in a bounded window around n^{-1/m2(H)}; if the theorem needs p bounded above by a constant multiple of n^{-1/m2(H)}, insert the standard restriction argument (take a random subgraph with probability p0/p) and recheck the constants in Claim 4.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I reviewed the proof chain from the deterministic template entropy through the sparse and dense transference arguments. The sparse lower bound in §4.1 correctly uses a 2-balanced subgraph attaining m2(H), and the variance calculation in Lemma 4.1 checks out. The dense upper bound in §4.2 is also coherent: the multicolour sparse regularity lemma supplies a common partition, Claim 4.2 correctly converts an SDR on the reduced template into a rainbow H via Lemma 2.2, and the certificate-counting bounds (4.2)–(4.5) have no hidden factor-of-two or parameter-dependency error. Proposition 3.2's double-counting, defective-injection gap, and mixing-lemma conclusion are internally consistent, including the endpoint subtlety of Proposition 2.5 that excludes the endpoint from the stability conclusion. The only genuinely externalized step is Lemma 2.2, but its use here is standard: with fixed M, parts of linear size, and density α, the KŁR theorem applies for all p ≥ A n^{-1/m2(H)}. I found no concrete failure, so I am not raising a substantive objection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies, for a fixed graph H with q=e(H)≥3 that is not a matching, and a fixed integer ℓ≥q, the exponential number R_{H,ℓ}(G(n,p)) of ℓ-edge-colourings of the binomial random graph with no rainbow copy of H. Theorem 1.4 states that below p≤a n^{-1/m_2(H)} the base ℓ survives on almost all edges, while above p≥A n^{-1/m_2(H)} the exponential rate is governed by a deterministic template-entropy constant λ(H,ℓ) up to δ in the exponent. Corollary 1.5 evaluates λ(H,ℓ)=log(q−1) throughout q≤ℓ≤(q−1)^{q/(q−2)}; Theorem 1.6 gives a counting-stability version strictly below the endpoint; Theorem 1.7 gives deletion-profile bounds and characterizes when the (q−1)-colour rate persists for every fixed ℓ. The proofs combine multicolour sparse regularity, the KŁR embedding theorem, Chernoff-type concentration, a local Hall-palette inequality, and a reduced-template entropy dichotomy.","tokens_in":16318,"tokens_out":8431,"duration_ms":87566,"significance":"If the results are correct, this is a substantial and timely contribution to the enumerative anti-Ramsey theory of random graphs. The paper extends the Gallai-colouring transition from triangles to every fixed non-matching graph and, more importantly, separates the random transference from the underlying dense palette optimisation: the deterministic rate λ(H,ℓ) is defined independently of any fitted parameter or hidden normalization, and the sparse and dense bounds are proved by standard, reproducible ingredients. The exact evaluation of λ(H,ℓ)=log(q−1) in a universal Hall range, the robust counting stability below the endpoint, and the deletion-profile bounds for large colour sets give a coherent and fairly complete picture. The only genuinely external input, Lemma 2.2, is stated explicitly and applied in a standard KŁR regime; I found no circular step or unsupported parameter dependence in the manuscript.","major_comments":[],"minor_comments":[{"comment":"In the proof of Proposition 3.2, the text says 'Theorems 2.4 and 2.5 give local entropy at most q log(q−1)'; these should be Lemma 2.4 and Proposition 2.5, respectively.","section":"§3.1"},{"comment":"Throughout §4, the statements of Lemma 2.1, Lemma 2.2, Lemma 2.3, Proposition 3.1, and Lemma 4.1 are cited as 'Theorem 2.1', 'Theorem 2.2', 'Theorem 2.3', 'Theorem 3.1', and 'Theorem 4.1'; please harmonize the cross-referencing with the actual labels.","section":"§4"},{"comment":"The sentence after Eq. (4.6) says 'Combining this theorem with Theorem 3.3 ... proves Theorem 1.5'; this should refer to Corollary 3.3 and Corollary 1.5, respectively.","section":"§4.2"},{"comment":"The displayed boundary values '25, 27, 14 4/3 ≈ 33.742' are typeset ambiguously; please write the exponent explicitly (for example 14^{4/3}) and state which deletion profile yields each boundary value.","section":"§5.3"}],"recommendation":"minor_revision","confidential_remarks":"No confidential concerns: the citation and attribution practices appear appropriate, and the reliance on the KŁR input is explicitly stated rather than hidden."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is a solid paper. It extends the Benevides–Monteiro–Mota Gallai transition to every fixed non-matching H and all fixed ℓ≥e(H), at the natural scale n^{-1/m2(H)}, and it splits the problem cleanly into a deterministic template-entropy optimization and a sparse random transference. I read the proof chain carefully, and it holds together.\n\nWhat's genuinely new: Theorem 1.4 is the first transference of this type beyond triangles; Corollary 1.5 evaluates the dense rate exactly (base q−1) on the whole interval q≤ℓ≤(q−1)^{q/(q−2)}; Theorem 1.6 gives a counting-stability statement below the endpoint; Theorem 1.7 gives deletion-profile bounds that nail the large-ℓ first-order behaviour and the case where the q−1 rate persists for all ℓ. The local Hall-palette inequality (Lemma 2.4) is a neat multiplicative form of Hall's obstruction, and the robust gap version is correct. The proof of Proposition 2.5 is fine (I checked the derivative). The sparse lower bound via a 2-balanced subgraph and the variance calculation in Lemma 4.1 are standard and sound. The dense upper bound uses multicolour sparse regularity plus KŁR in the standard uniform form; Claim 4.2 is exactly where the rainbow obstruction is transferred, and it works. The stability argument with the mixing lemma (Lemma 2.6) is clear.\n\nSoft spots, in proportion. The main external load-bearing input is the uniform KŁR-type embedding lemma (Lemma 2.2), quoted from Conlon–Gowers–Samotij–Schacht. That is a heavy theorem, but it is used in a completely standard regime—fixed H, parts of linear size, density α—and I don't see any hidden parameter problem. The transference is one-sided in the sense that the exact rate is only captured for p≥A n^{-1/m2(H)}; the constant-factor window around the threshold is not addressed, which is normal for this method. For ℓ beyond the Hall range the deterministic optimisation is not solved exactly; the deletion-profile bounds give only first-order behaviour, and the paper says so openly (Section 5.4). That is a separate dense extremal problem, not a gap in the random argument. A minor point: the statement of Lemma 2.2 bundles the KŁR constants; the proof delegates to the standard trimming argument, so the uniformity across all choices of U_i, J_xy is assumed. Acceptable, but a referee should check the constant chasing in Claim 4.2 and equations (4.2)–(4.4) once carefully. I did not find an error.\n\nBottom line: this is a strong paper. The reader's ACCEPT verdict is right. It deserves a serious referee, and I expect it to survive. I would cite it and bring it to our reading group—it is a clean example of entropy transference.\n\nSend it out.","headline":"A complete and correct proof of the rainbow-H-free counting transition for every fixed non-matching H; the deterministic-entropy separation does real work.","tokens_in":16892,"tokens_out":3599,"would_cite":true,"duration_ms":34485,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C15","05C35","05C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a transference principle: the exponential number of rainbow-H-free edge-colourings of G(n,p) is governed by a deterministic template-entropy optimisation above the threshold p=n^{-1/m2(H)}, and the rate equals…","keywords":["rainbow-free colourings","random graphs","template entropy","sparse regularity","anti-Ramsey theory","Erdős–Rothschild problem","maximum 2-density","Hall's theorem"],"falsifier":"For H=K_4 and ℓ=6, which lies inside the claimed Hall range, one could search over all palette assignments on K_n that give no system of distinct representatives on any K_4 and check whether any assignment for small n has average log-palette size exceeding log 5. If such an assignment exists, Corollary 1.5 would fail; the paper predicts none does. Alternatively, one could compute R_{H,ℓ}(G(n,p)) numerically for moderate n and p above the threshold and compare the empirical exponent with λ(H,ℓ).","tokens_in":15925,"feed_emoji":"🎨","tokens_out":6103,"duration_ms":64148,"temperature":0.7,"pith_summary":"The paper determines, up to a subexponential factor, how many edge-colourings of the binomial random graph G(n,p) avoid a rainbow copy of a fixed graph H, for any fixed number of colours. It proves an entropy-transference theorem at the natural scale p=$n^{{-1/m2(H)}}$: below a small constant multiple of this scale almost all edges can be coloured freely, while above a large constant multiple the exponential counting rate is set by a deterministic optimisation over complete graphs. For every colour count up to a universal endpoint, that rate is exactly (e(H)-1)^{e(G)}. This extends the known triangle transition to every fixed non-matching graph and separates the sparse random transfer from the dense palette optimisation.","feed_headline":"Rainbow-free colouring counts pinned for every fixed graph","feed_subtitle":"Above the threshold the rate is deterministic template entropy; in a universal colour range it is exactly (q-1)^{e(G)}.","key_machinery":"The load-bearing object is the H-admissible palette template on K_n: an assignment of a nonempty colour set to each edge such that the q edge-palettes on every copy of H have no system of distinct representatives. Its entropy is Ent(P)=Σ_e log|P(e)|, and λ(H,ℓ) is the limiting maximal normalised entropy. The key local identity is the Hall-palette inequality: for q nonempty palettes with no SDR, the product of their sizes is at most max_{2≤s≤q} (s-1)^s $ℓ^{{q-s}}$, and this is at most (q-1)^q exactly when ℓ ≤ (q-1)^{q/(q-2)}, with equality forcing all q palettes to be a common (q-1)-set below the endpoint. Sparse regularity together with a uniform embedding lemma transfers this deterministic entropy bound from reduced templates to G(n,p), while a first-moment argument handles the sparse side.","core_discovery":"The central claim is a two-sided exponential description of R_{H,ℓ}(G(n,p)), the number of rainbow-H-free ℓ-colourings. For every fixed ℓ≥e(H) and every δ>0, with high probability R_{H,ℓ}(G(n,p)) ≥ $ℓ^{{(1-δ)e(G)}}$ when p ≤ a $n^{{-1/m2(H)}}$, and exp((λ(H,ℓ)-δ)e(G)) ≤ R_{H,ℓ}(G(n,p)) ≤ exp((λ(H,ℓ)+δ)e(G)) when p ≥ A $n^{{-1/m2(H)}}$. Here λ(H,ℓ) is the limiting maximum average logarithmic palette size over all H-admissible templates on complete graphs, that is, palette assignments in which the edges of every copy of H admit no system of distinct representatives. The paper evaluates this rate throughout the range q ≤ ℓ ≤ (q-1)^{q/(q-2)} as λ(H,ℓ) = log(q-1), where q=e(H), and proves a counting-stability version below the endpoint: colourings that stay far from using only q-1 colours have exponentially smaller count.","pith_inferences":["Editorial inference: if the transference principle is correct, then any future exact evaluation of λ(H,ℓ) for a particular H automatically gives the sparse random counting rate, so the random problem is fully reduced to a deterministic dense optimisation.","Editorial inference: the paper leaves open whether deletion-profile templates exhaust all obstructions to optimality of the constant (q-1)-palette; a natural testable extension is to compute λ(H,ℓ) for specific pairs such as H=K_4 and ℓ between 7 and 11 to see where the constant-palette rate first fails.","Editorial inference: the counting-stability theorem suggests a typical-structure description of random rainbow-H-free colourings above the threshold: almost all such colourings either use q-1 colours almost everywhere or pay an exponential entropy penalty, which could support sampling or decomposition algorithms."],"forward_implications":["For every fixed non-matching H and every colour count ℓ with e(H) ≤ ℓ ≤ (e(H)-1)^{e(H)/(e(H)-2)}, the dense-side exponential rate of rainbow-H-free colourings of G(n,p) is (e(H)-1)^{e(G)}.","For H=K_3 and ℓ=3 the theorem recovers the exponential form of the previously known triangle transition.","If H-e is bipartite for some edge e, then for every fixed ℓ the dense-side rate is log(q-1) per edge, no matter how many colours are allowed.","For large colour sets, the first-order behaviour is governed by edge-deletion profiles: λ(H,ℓ)/log ℓ tends to the Turán density of the family of one-edge deletions of H.","Below the Hall endpoint the colourings are counting-stable: any colouring that requires more than ε e(G) recolourings to reduce to q-1 colours occurs with exponentially smaller count."],"supporting_citations":[{"why":"Quoted as Lemma 2.2; supplies the uniform sparse embedding lemma that turns a system of distinct representatives in a reduced template into a real rainbow copy of H in G(n,p).","marker":"[7]"},{"why":"Establishes the random-host Gallai transition for triangles, which this paper generalises to every fixed non-matching H and every fixed number of colours.","marker":"[6]"},{"why":"Provides the colouring-template entropy formalism used to define λ(H,ℓ) and to set up the deterministic optimisation.","marker":"[11]"},{"why":"Hall's theorem underlies the local Hall-palette inequality of Lemma 2.4.","marker":"[17]"},{"why":"Supplies the multicolour sparse regularity lemma used to build reduced certificates in the dense transference argument.","marker":"[13]"},{"why":"Used with the deletion construction to bound the entropy of complete graphs through the extremal density of the one-edge-deleted family H^-.","marker":"[21]"}],"fun_headline_variants":["Entropy transference gives exact rainbow-free colouring counts","Rainbow-H-free counts pinned exactly in universal range","Exact rate for rainbow-H-free colourings in random graphs","Rainbow-free counting transition from triangles to all graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The dense upper bound relies on a quoted sparse-embedding lemma asserting that, once p is a large constant multiple of $n^{{-1/m2(H)}}$, any collection of regular cluster pairs of positive density arranged like H must contain a transversal copy of H; if that lemma fails for the small density parameter α<1/(3ℓ) or for the partition bound M used in the proof, the upper estimate does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Entropy transference gives exact rainbow-free colouring counts","Rainbow-H-free counts pinned exactly in universal range","Exact rate for rainbow-H-free colourings in random graphs","Rainbow-free counting transition from triangles to all graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000897,"raw_usage":{"total_tokens":3890,"prompt_tokens":1000,"completion_tokens":2890,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":616,"completion_tokens_details":{"reasoning_tokens":2824}},"tokens_in":616,"tokens_out":2890,"duration_ms":24577,"temperature":1.0,"reasoning_tokens":2824,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:11:23.820233+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For H=K_4 and ℓ=6, which lies inside the claimed Hall range, one could search over all palette assignments on K_n that give no system of distinct representatives on any K_4 and check whether any assignment for small n has average log-palette size exceeding log 5. If such an assignment exists, Corollary 1.5 would fail; the paper predicts none does. Alternatively, one could compute R_{H,ℓ}(G(n,p)) numerically for moderate n and p above the threshold and compare the empirical exponent with λ(H,ℓ).","supporting_citations":[{"cited_title":"Conlon, W","cited_arxiv_id":null,"evidence_quote":"Quoted as Lemma 2.2; supplies the uniform sparse embedding lemma that turns a system of distinct representatives in a reduced template into a real rainbow copy of H in G(n,p)."},{"cited_title":"Gallai 3-colourings of random graphs","cited_arxiv_id":"2604.04115","evidence_quote":"Establishes the random-host Gallai transition for triangles, which this paper generalises to every fixed non-matching H and every fixed number of colours."},{"cited_title":"Falgas-Ravry, K","cited_arxiv_id":null,"evidence_quote":"Provides the colouring-template entropy formalism used to define λ(H,ℓ) and to set up the deterministic optimisation."},{"cited_title":"Hall,On representatives of subsets, J","cited_arxiv_id":null,"evidence_quote":"Hall's theorem underlies the local Hall-palette inequality of Lemma 2.4."},{"cited_title":"Gerke and A","cited_arxiv_id":null,"evidence_quote":"Supplies the multicolour sparse regularity lemma used to build reduced certificates in the dense transference argument."},{"cited_title":"Simonovits,A method for solving extremal problems in graph theory, stability problems, in Theory of Graphs (Proc","cited_arxiv_id":null,"evidence_quote":"Used with the deletion construction to bound the entropy of complete graphs through the extremal density of the one-edge-deleted family H^-."}],"review_version":1}