{"id":"4dc5f397-f6e1-4ffa-9d8a-7eecf4ff6cf5","arxiv_id":"2607.03035","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Blowup thresholds are always positive for non-bipartite H, fail monotonicity under induced subgraphs, and equal 1/4 for certain constrained odd-cycle blowups.","lead":"The paper shows that the blowup threshold of a forbidden graph H is always positive for non-bipartite H, is not monotone under induced subgraphs, and equals 1/4 for a family of constrained odd-cycle blowups. This separates a rigid exact-template notion of structure from the classical chromatic threshold.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The Reader correctly flags the external stability lemma as the only non-elementary black box, but that lemma is used solely to obtain an almost-bipartite partition; the paper’s own argument then upgrades it to an exact complete bipartite graph under maximality and the minimum-degree hypothesis. Because the upgrade step is independent of the precise quantitative form of the stability statement, a hypothetical failure of Łuczak–Simonovits for the constrained family would not by itself invalidate δ_B(H) ≤ 1/4. The positivity and non-monotonicity theorems rely only on the authors’ own pseudo-blowup constructions and case analysis, both of which appear free of gaps. Consequently the Reader’s ACCEPT verdict stands; the single concrete check proposed is a low-cost sanity verification of the shared reduction lemma rather than a challenge to the central claims.","tokens_in":24690,"tokens_out":465,"duration_ms":4095,"concrete_test":"Independently verify the pigeonhole extraction in Lemma 2.2 (the only non-trivial reduction used for all lower bounds) on a small explicit instance: take M = C_5 with one pendant edge, n2 = 20, C = 3, t = 2; confirm that any blowup of a 3-vertex graph containing the corresponding (n1,n2)-pseudo-blowup must contain M[2]. If the extraction fails, the positivity and 1/4 lower bounds would need re-examination.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The three main theorems rest on self-contained combinatorial constructions (pseudo-blowups for lower bounds, explicit case analysis for non-monotonicity) and a standard external stability lemma (Łuczak–Simonovits) that is correctly applied only as a starting point for the upper bound. The subsequent promotion from almost-bipartite to exact bipartite (Claim 4.3 and the thickening argument) is elementary and does not inherit any hidden failure mode of the stability statement for the constrained family C^{≥4}_{2s-1}. No internal inconsistency, circularity, or unstated assumption that would collapse Theorems 1.1–1.3 was found.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the blowup threshold δ_B(H), which asks for the minimum-degree density forcing every maximal H-free graph to be an exact blowup of a bounded template graph (a rigid strengthening of the chromatic and homomorphism thresholds). It proves three main results: δ_B(H) > 0 for every non-bipartite H (Theorem 1.1), that δ_B is not monotone under induced subgraphs (Theorem 1.2, via an explicit H with δ_B(H) ≥ 1/2 > δ_B(H ∪ K_3)), and that δ_B(H) = 1/4 exactly for the natural family of 3-chromatic constrained odd-cycle blowups C^4_{2s-1} (Theorem 1.3, obtained by matching the lower-bound pseudo-blowup construction of Theorem 1.5 with the upper bound of Theorem 1.4).","tokens_in":24867,"tokens_out":998,"duration_ms":14908,"significance":"If correct, the results cleanly separate the blowup threshold from the chromatic-threshold spectrum: positivity never vanishes outside the bipartite world (in contrast to δ_χ(C_{2k-1}) = 0), non-monotonicity under induced subgraphs blocks any direct transfer of the Allen–Böttcher–Griffiths–Kohayakawa–Morris classification strategy, and the exact value 1/4 lies outside {0, 1/3, 1/2}. The pseudo-blowup framework (Definition 2.1 + Lemma 2.2/Corollary 2.3) supplies a reusable, concrete lower-bound engine, while the upper-bound arguments upgrade standard almost-bipartite stability (via maximality and elementary thickening) to exact blowups. These are solid, self-contained contributions that enlarge the structural landscape of dense H-free graphs.","major_comments":[],"minor_comments":[{"comment":"Definition 2.1: the three admissible sparse rules (i = j, i < j, i > j) are clear, but a one-sentence remark that the rule is fixed once and for all for each small-block pair would prevent any ambiguity when the same model is reused for different H.","section":"Section 2"},{"comment":"Lemma 2.5 / Claim 2.6: the argument that a non-singular vertex forces a second neighbour inside a matching pair is correct, yet the subsequent distance calculation on the projected cycle (gaps of length at least 2 except one) could be illustrated with a short diagram or an explicit listing of the 4k-3 lower bound for the singleton interval.","section":"Section 2.1"},{"comment":"Section 3.2: the constant 58 arising from the dense-matching number is harmless for the existence proof, but a parenthetical remark that any fixed bound works (and that the precise value is irrelevant) would reassure the reader that no optimisation is claimed.","section":"Section 3.2"},{"comment":"Figure 4.1 and the surrounding text: the distinction between C^ℓ_ℓ and C^≥ℓ_ℓ is visually clear, yet a single sentence defining “singleton interval of length exactly ℓ with all other parts size ≥ 2” in the caption itself would make the figure self-contained.","section":"Section 4"},{"comment":"Throughout: the notation Fr·s for an arbitrary blowup appears with varying typography (Fr·s, F[r], etc.); a uniform choice (e.g., F[r]) would improve readability.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a natural and clean follow-up to the authors’ earlier introduction of the blowup threshold. The reliance on the Łuczak–Simonovits stability lemma is standard and correctly limited to a starting point; the subsequent elementary promotion to an exact blowup does not inherit any potential failure mode for the constrained family. Fit for a strong combinatorics journal is excellent."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper cleanly shows that the blowup threshold behaves differently from the chromatic one in three concrete ways: it is always positive for non-bipartite H, it fails monotonicity under induced subgraphs, and it hits the exact value 1/4 for a natural family of constrained odd-cycle blowups. Those three statements are new relative to Allen–Böttcher–Griffiths–Kohayakawa–Morris, Thomassen, Sankar, and the authors’ own earlier introduction of δ_B.\n\nThe lower bounds rest on an explicit pseudo-blowup gadget (large blocks complete, selected small-block pairs matched or ordered). Lemma 2.2 and Corollary 2.3 turn any such dense H-free pseudo-blowup into a genuine lower bound; the constructions for positivity and for the 1/4 family are short and checkable. Non-monotonicity is proved by a concrete 6-vertex H with δ_B ≥ 1/2 and a careful case analysis showing that adding a disjoint triangle drops the threshold strictly below 1/2 (exceptional set of size ≤58, then maximality upgrades to a bounded blowup). The upper bound for 1/4 starts from the standard Łuczak–Simonovits almost-bipartite stability and then uses the minimum-degree condition to thicken any remaining odd cycle into a forbidden constrained blowup; once the graph is bipartite, maximality forces a complete bipartite graph. The stress-test note is right: the promotion step does not inherit any hidden failure mode of the stability lemma for the constrained family.\n\nSoft spots are minor. The non-monotonicity case analysis is long and a bit tedious, and the exact value is only for one family, so the full 3-chromatic spectrum remains open. Citations are appropriate; self-citations are only for context. No free parameters, no circularity.\n\nThis is for people working on structural extremal thresholds (chromatic, homomorphism, VC, blowup). It supplies a concrete new benchmark and a reusable construction technique. I would send it to a serious referee without hesitation.","headline":"Solid separation of blowup thresholds from chromatic ones: positivity, non-monotonicity, and a clean exact value 1/4.","tokens_in":25456,"tokens_out":521,"would_cite":true,"duration_ms":4754,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C15","05C75"],"pacs":[],"model":"grok-4.5","headline":"Dense maximal H-free graphs become exact blowups of a bounded template once the minimum degree clears a positive threshold that never vanishes for non-bipartite H.","keywords":["blowup threshold","chromatic threshold","homomorphism threshold","maximal H-free graphs","pseudo-blowups","odd-cycle blowups","minimum degree","twin quotient"],"falsifier":"Exhibit a single 3-chromatic constrained blowup H of an odd cycle, together with an infinite family of maximal H-free graphs of minimum degree strictly larger than n/4 whose twin quotients remain unbounded.","tokens_in":25610,"feed_emoji":"▦","tokens_out":705,"duration_ms":5150,"temperature":0.7,"pith_summary":"The paper studies the blowup threshold δ_B(H): the minimum-degree density above which every maximal H-free graph must be an exact blowup of some bounded template graph. This is stricter than the classical chromatic and homomorphism thresholds, which only guarantee bounded colouring or a bounded H-free quotient. The authors prove three structural facts that separate δ_B from those weaker parameters. First, δ_B(H) is always strictly positive whenever H is non-bipartite; the threshold therefore never collapses to zero outside the bipartite world. Second, the parameter is not monotone under induced subgraphs, so global features of H matter and the usual reduction to minimal forbidden subgraphs fails. Third, for a natural family of 3-chromatic constrained blowups of odd cycles one obtains the exact value δ_B(H)=1/4, a density that is impossible for the chromatic threshold. Taken together, the results show that upgrading approximate or quotient structure to genuine homogeneous blowups produces a richer and more delicate spectrum.","feed_headline":"Blowup thresholds stay positive for every non-bipartite H","feed_subtitle":"Dense maximal H-free graphs become exact bounded blowups; the spectrum is richer than chromatic thresholds","key_machinery":"The pseudo-blowup construction: replace vertices of a model graph by large and small blocks, keep most adjacent pairs complete bipartite, but replace selected small-block pairs by sparse ordered patterns (matchings or strict inequalities). Sparse pairs force the twin quotient to be unbounded, while the minimum-degree condition still yields a concrete lower bound on δ_B.","core_discovery":"For every non-bipartite graph H the blowup threshold satisfies δ_B(H)>0; the same threshold is not monotone under induced subgraphs; and there exists a natural family of 3-chromatic constrained odd-cycle blowups for which δ_B(H) equals exactly 1/4.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Blowup thresholds stay positive for all non-bipartite H","δ_B never vanishes outside bipartite graphs","Blowup threshold not monotone under induced subgraphs","Exact δ_B=1/4 for constrained odd-cycle blowups","Blowup thresholds diverge from chromatic spectrum"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The exact upper bound of 1/4 rests on an external stability theorem that every sufficiently dense graph forbidding the relevant odd-cycle blowups becomes almost bipartite after deleting a negligible number of edges.","fun_headline_variants_meta":{"raw":{"variants":["Blowup thresholds stay positive for all non-bipartite H","δ_B never vanishes outside bipartite graphs","Blowup threshold not monotone under induced subgraphs","Exact δ_B=1/4 for constrained odd-cycle blowups","Blowup thresholds diverge from chromatic spectrum"]},"model":"grok-4.5","effort":"low","cost_usd":0.004576,"raw_usage":{"total_tokens":1376,"prompt_tokens":824,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":45760000,"prompt_tokens_details":{"text_tokens":824,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":491,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":824,"tokens_out":61,"duration_ms":3538,"temperature":1.0,"reasoning_tokens":491,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T05:17:53.651536+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a single 3-chromatic constrained blowup H of an odd cycle, together with an infinite family of maximal H-free graphs of minimum degree strictly larger than n/4 whose twin quotients remain unbounded.","supporting_citations":[],"review_version":1}