{"id":"6d9ef322-1ebb-43ce-98d2-2fc0ef13a521","arxiv_id":"2508.07533","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For sufficiently large n, the exact Turan number of the generalized book graph B_{r,k} is found in non-r-partite graphs, together with all extremal graphs, for every r>=3 and k>=1.","lead":"This paper determines the largest number of edges in a large graph that avoids a book-shaped structure B_{r,k} while also not being r-partite. It also identifies every graph that achieves this maximum.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified from the abstract; exact claim is plausible but not auditable without the proof.","rationale":"The reader's verdict is UNVERDICTED because only the abstract is available. I agree: the paper's claim is plausible and consistent with the known framework of stability and local structure for color-critical graphs, but no proof text exists to audit. I could not identify a specific internal inconsistency or a concrete missing extremal configuration from the abstract alone. The weakest point is indeed the completeness of the local structure characterization, but that is a verification gap rather than a demonstrated flaw. Therefore I do not propose changing the reader's verdict; the appropriate disposition remains UNVERDICTED until the full proof is supplied.","tokens_in":923,"tokens_out":25820,"duration_ms":314420,"concrete_test":"Obtain the full manuscript and independently re-derive the local structure theorem from its stability lemma, checking that the classification covers all r>=3 and all k>=1, including the separate k=1 case where B_{r,1}=K_{r+1}. Concretely, for r=3,k=2, run an exhaustive computer search over all non-4-partite B_{3,2}-free graphs on n=9 and compare the maximum edge count with the theorem's formula evaluated at n=9; a mismatch would indicate the classification is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is an exact extremal theorem whose truth hinges on a complete local-structure classification of near-extremal non-r-partite B_{r,k}-free graphs. The abstract states the result but gives neither the extremal formula nor the list of extremal graphs, so there is no specific assertion I can check for a hidden counterexample. I therefore do not assert a mathematical flaw. The single most load-bearing unverified step is the completeness of that local classification: if any non-r-partite B_{r,k}-free configuration with higher edge count is missing, both the exact value and the extremal-family identification would fail. This is a genuine verification gap, not a demonstrated error, and it is exactly the reason the result should remain UNVERDICTED until the full proof is available.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper (arXiv:2508.07533, math.CO) studies the Turán problem for generalized book graphs B_{r,k} in non-r-partite graphs. B_{r,k} is obtained by joining every vertex of a K_r to every vertex of an independent set of size k; it is color-critical with chromatic number r+1. For r >= 3, k >= 1, and sufficiently large n, the paper claims to determine exactly the maximum number of edges in a non-r-partite B_{r,k}-free graph on n vertices, denoted ex_{r+1}(n,B_{r,k}), and to identify all extremal graphs. The abstract states that the proof is based on stability theory and a local structure characterization of near-extremal non-r-partite B_{r,k}-free graphs. No full text, theorem statement with the extremal formula, or proof details are available for review.","tokens_in":1092,"tokens_out":1685,"duration_ms":18203,"significance":"If the claimed result is correct, it constitutes a substantial contribution to extremal graph theory: it extends Simonovits' classical color-critical Turán theorem to the non-r-partite setting for the family of generalized book graphs. Exact extremal numbers and extremal structures for non-r-partite H-free graphs are known only for very few color-critical H, so an exact result for all r >= 3 and k >= 1 would be significant. The paper also promises a complete local structure classification, which, if valid, would be a valuable technical tool. However, because the full manuscript is not available, the significance cannot be fully assessed from the abstract alone; the central claim is plausible but not auditable.","major_comments":[{"comment":"The central claim—the exact value of ex_{r+1}(n,B_{r,k}) and the identification of Ex_{r+1}(n,B_{r,k})—is stated only qualitatively. The abstract does not provide the extremal formula, the list of extremal graphs, or the threshold on n. This prevents any independent check of the theorem. A referee cannot verify that the stated result is internally consistent without the missing details.","section":"Abstract"},{"comment":"The proof is said to rely on a 'local structure characterization' of near-extremal non-r-partite B_{r,k}-free graphs. The completeness of this classification is load-bearing: if any near-extremal configuration is omitted, both the exact extremal number and the extremal-family identification would change. The abstract gives no indication of how this classification is established, how exhaustiveness is proven, or what the resulting extremal family is. This is a verification gap, not a demonstrated error, but it is the central point on which the paper's correctness rests.","section":"Abstract"},{"comment":"The paper positions itself as an exact result ('the exact value') but does not state the value in the abstract. In particular, it is unclear whether the extremal graphs are unique or form a small family, and whether the extremal number is a simple closed form or a case-dependent expression. Without this information, the claimed theorem is not meaningfully testable from the available text.","section":"Abstract"}],"minor_comments":[{"comment":"The notation Ex_{r+1}(n,H) is used but the defining condition (non-r-partite, H-free, size exactly ex_{r+1}(n,H)) is given only in prose. For the abstract, a concise formal definition would improve clarity.","section":"Abstract"},{"comment":"The phrase 'based on the stability theory and local structure characterization' is vague; a short indication of the main steps (e.g., stability, then local classification, then exact extremal construction) would help the reader understand the methodology.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only review; the full manuscript was not provided. The claim is plausible and fits the journal's scope, but the central load-bearing element—the local structure classification—is completely unverified. I cannot recommend acceptance without seeing the actual theorem statement and proof. I suggest requesting the full text for a proper review. No concerns about authorship or citation practices can be assessed from the abstract."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Your abstract-only read matches mine. The paper states a clean, exact result: for every r≥3 and k≥1, once n is large enough, the maximum edges in a non-r-partite B_{r,k}-free graph is pinned down, and every extremal graph is identified. That is a genuine extension of Simonovits's color-critical theorem, which only tells you about the r-partite extremal graphs. The non-r-partite regime is where the extremal structure is richer, so this is a solid contribution if the proof is correct.\n\nWhat looks good from the abstract: the problem is well posed, the claimed theorem is specific, and the announced route—stability plus local structure classification—is a sensible, established program for this kind of problem. I see no circularity: the Turán number is defined independently, and stability and local structure are external tools. The citation pattern is to the standard theorem; nothing suspicious from here.\n\nWhere the soft spots are: the abstract gives no extremal formula and no list of extremal graphs, and it gives no threshold on n. That means the one step that carries all the weight—the completeness of the local structure characterization of near-extremal non-r-partite B_{r,k}-free graphs—is completely unverified. If a single near-extremal configuration is missing, both the exact value and the extremal family would be wrong. That's not a demonstrated flaw; it's a genuine verification gap, and it's exactly why this should not be accepted on faith. Also, the stability step itself is not shown, but that's standard.\n\nOne minor point: the paper breaks no new methodological ground; the strategy is the standard stability pipeline. That doesn't diminish the result, but it means the value is in the execution and the completeness of the case analysis, not in the novelty of the approach.\n\nBottom line: if the local structure argument is complete, this is a worthwhile paper that sharpens a classical theorem. But from the abstract alone I can't tell. It absolutely deserves to go to a serious referee, and the referee should be told to focus on the classification lemma for near-extremal graphs. I'd want to see the full proof before citing it, and I'd bring it to reading group once the full text is available.","headline":"Plausible and precise exact extremal result for generalized books in the non-r-partite regime, but the abstract alone leaves the load-bearing local classification unverified.","tokens_in":1532,"tokens_out":2441,"would_cite":false,"duration_ms":25761,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper determines the exact Turán number for non-r-partite graphs that avoid the generalized book $B_{r,k}$, for large $n$.","keywords":["Turán number","generalized book graph","non-r-partite graphs","color-critical graphs","stability method","extremal graphs","extremal graph theory"],"falsifier":"Find any $r\\ge 3$, $k\\ge 1$ and arbitrarily large $n$ for which a non-$r$-partite $B_{r,k}$-free graph on $n$ vertices has more than the paper's $\\mathrm{ex}_{r+1}(n,B_{r,k})$ edges, or an extremal graph that is not in the identified family. Either would refute the exactness claim.","tokens_in":853,"feed_emoji":"📐","tokens_out":6626,"duration_ms":60320,"temperature":0.7,"pith_summary":"For a color-critical graph $H$ with chromatic number $r+1$, Simonovits' theorem fixes the extremal $H$-free graph for large $n$: the $r$-partite Turán graph $T_r(n)$. This paper considers the companion problem for the generalized book $B_{r,k}$ (a $K_r$ with an independent set of $k$ vertices joined to all of it): among $n$-vertex graphs that are not $r$-partite and contain no $B_{r,k}$, which one has the most edges? The paper asserts that for $r\\ge 3$, $k\\ge 1$ and all sufficiently large $n$, this maximum is exactly determined, and every graph attaining it is explicitly identified. The proof route is stability theory followed by a local structure characterization. If the characterization is exhaustive, the non-$r$-partite Turán problem for books is completely solved rather than approximated.","feed_headline":"Exact edge maximum found for non-r-partite book-free graphs","feed_subtitle":"For every large non-r-partite graph avoiding the book B_{r,k}, the largest edge count and every extremal example are now pinned down.","key_machinery":"The central object is the generalized book $B_{r,k}$, and the main quantity is $\\mathrm{ex}_{r+1}(n,B_{r,k})$, the maximum number of edges in a non-$r$-partite $B_{r,k}$-free graph. The argument's engine is the pair 'stability theory and local structure characterization': stability theory shows that any near-extremal graph is structurally close to the $r$-partite Turán graph $T_r(n)$, and the local structure characterization then classifies all near-extremal non-$r$-partite configurations tightly enough to extract the exact maximum and the exact extremal family.","core_discovery":"The paper's contribution is the exact value of $\\mathrm{ex}_{r+1}(n,B_{r,k})$ for $r\\ge 3$, $k\\ge 1$ and sufficiently large $n$, together with the complete description of $\\mathrm{Ex}_{r+1}(n,B_{r,k})$, the set of edge-maximal non-$r$-partite $B_{r,k}$-free graphs on $n$ vertices. Here $B_{r,k}$ is the graph obtained by joining every vertex of a $K_r$ to every vertex of an independent set of size $k$; it has chromatic number $r+1$ and is color-critical. The claimed result is the exact, not asymptotic, counterpart to Simonovits' theorem at the next chromatic level: no graph outside the identified family can tie or exceed the stated maximum.","pith_inferences":["The same stability-plus-local-structure strategy is likely to yield exact non-$r$-partite Turán numbers for other color-critical graphs of chromatic number $r+1$, not just books; the paper does not claim this extension.","A natural follow-up is the minimum $n$ from which the exact formula holds; the paper only says 'sufficiently large', and the threshold presumably depends on both $r$ and $k$.","The abstract does not describe the extremal family in words; a likely shape is the Turán graph $T_r(n)$ with a small book-free modification (for instance one extra edge inside a part), but that specific form is my inference from the classical pattern, not a statement made here."],"forward_implications":["For all sufficiently large $n$, the number $\\mathrm{ex}_{r+1}(n,B_{r,k})$ is known exactly, so asymptotic analysis for book-free non-$r$-partite graphs is replaced by a precise formula.","The set $\\mathrm{Ex}_{r+1}(n,B_{r,k})$ is explicit, so an edge-maximal non-$r$-partite $B_{r,k}$-free graph can be checked against a finite list rather than characterized only up to error terms.","Together with Simonovits' theorem for $r$-partite extremal graphs, the maximum edge count is now resolved on both sides of the chromatic threshold: within $r$-partite graphs and just above them.","The result gives a complete answer for every $r\\ge 3$ and $k\\ge 1$, not for a single small instance, which is what an exact theorem buys."],"supporting_citations":[],"fun_headline_variants":["Exact edge optimum for non-r-partite book-free graphs","Non-r-partite book-free graphs: exact edge maximum","Precise Turán number for non-r-partite book-free graphs","Book-free non-r-partite graphs: extremal count finalized"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The result stands or falls on the completeness of the local structure characterization: if some near-extremal non-$r$-partite $B_{r,k}$-free graph is not covered by the classification, the claimed exact maximum or the listed extremal graphs could be wrong.","fun_headline_variants_meta":{"raw":{"variants":["Exact edge optimum for non-r-partite book-free graphs","Non-r-partite book-free graphs: exact edge maximum","Precise Turán number for non-r-partite book-free graphs","Book-free non-r-partite graphs: extremal count finalized"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000222,"raw_usage":{"total_tokens":1417,"prompt_tokens":995,"completion_tokens":422,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":739,"completion_tokens_details":{"reasoning_tokens":350}},"tokens_in":739,"tokens_out":422,"duration_ms":4715,"temperature":1.0,"reasoning_tokens":350,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T22:01:13.224409+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find any $r\\ge 3$, $k\\ge 1$ and arbitrarily large $n$ for which a non-$r$-partite $B_{r,k}$-free graph on $n$ vertices has more than the paper's $\\mathrm{ex}_{r+1}(n,B_{r,k})$ edges, or an extremal graph that is not in the identified family. Either would refute the exactness claim.","supporting_citations":[],"review_version":1}