REVIEW 3 major objections 2 minor 1 cited by
The exact Tur\'an number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs
T0 review · 3 major / 2 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read The paper determines the exact Turán number for non-r-partite graphs that avoid the generalized book $B_{r,k}$, for large $n$.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (3)
- [Abstract] 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.
- [Abstract] 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.
- [Abstract] 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.
minor comments (2)
- [Abstract] 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.
- [Abstract] 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.
Circularity Check
No circularity detectable from the abstract; the result is an independent extremal theorem with no visible reduction to its own inputs.
full rationale
The review is abstract-only, so the derivation chain cannot be walked in detail. Within the abstract, the target quantity ex_{r+1}(n,B_{r,k}) is defined independently of the proof methods: it is the maximum number of edges in non-r-partite B_{r,k}-free graphs. The claimed proof uses 'stability theory and local structure characterization,' which are standard external tools, not quantities defined in terms of the extremal number itself. No fitted parameters, no self-citations, and no uniqueness claims from the authors' prior work appear. There is hence no specific equation or definition by which the conclusion reduces to an input. The completeness of the local structure classification is a verification gap, not circularity. A non-finding with score 0 is appropriate because no circular step can be quoted.
Assumptions & free parameters
assumptions (3)
- standard math Turan theorem and Simonovits stability theorem for color-critical graphs
- domain assumption The local structure characterization exhausts all possible near-extremal non-r-partite B_{r,k}-free graphs
- domain assumption The threshold on n is finite and the asymptotic stability analysis becomes exact for large n
Cite this review
Pith. "Pith review of The exact Tur\'an number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs." pith.science (2026). https://pith.science/paper/M5CFOBEO
@misc{pith2026250807533,
author = {Pith},
title = {Pith review of: The exact Tur\'an number of generalized book graph $B_r,k$ in non-$r$-partite graphs},
year = {2026},
howpublished = {\url{https://pith.science/paper/M5CFOBEO}},
note = {Machine review of arXiv:2508.07533}
}
abstract
Given a graph $H,$ we say that a graph is \textit{$H$-free} if it does not contain $H$ as a subgraph. The Tur\'an number $\ex(n,H)$ of $H$ is the maximum number of edges in an $n$-vertex $H$-free graph, the set of all the corresponding extremal graphs is denoted by $\Ex(n, H)$. The study of Tur\'an number of graphs is a central topic in extremal graph theory. A graph is \textit{color-critical} if it contains an edge whose deletion reduces its chromatic number. Simonovits showed that if $H$ is a color-critical graph of chromatic number $r+1,$ then for sufficiently large $n,$ $\Ex(n, H)=\{T_r(n)\},$ the $r$-partite Tur\'an graph of order $n.$ Given a color-critical graph $H$ with chromatic number $r+1,$ it is interesting to determine $H$-free non-$r$-partite graphs with maximum number of edges. For a graph $H$ with chromatic number $r+1,$ denote $\ex_{r+1}(n,H)$ the maximum number of edges in non-$r$-partite $H$-free graphs of order $n,$ the set of all non-$r$-partite $H$-free graphs of order $n$ and size $\ex_{r+1}(n,H)$ is denoted by $\Ex_{r+1}(n, H)$. For $r\geq 3,\,k\geq1,$ the generalized book graph \({B}_{r,k}\) is a graph obtained by joining every vertex of $K_r$ to every vertex of an independent set of size \(k\). Note that \({B}_{r,k}\) is a color-critical graph of chromatic number $r+1.$ In this paper, based on the stability theory and local structure characterization, the exact value of $\ex_{r+1}(n,B_{r,k})$ is determined and all the corresponding extremal graphs are identified, where $r\geq 3,\,k\geq1$ and $n$ is sufficiently large.
Forward citations
Cited by 1 Pith paper
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Spectral Tur\'an-type problem in non-$r$-partite graphs: Forbidden generalized book graph $B_{r,k}$
For r>=3, k>=1 and sufficiently large n, the unique spectral extremal graph among non-r-partite B_{r,k}-free graphs is Y_r(n), obtained from the Turán graph T_r(n) by one internal edge and some cross-edge deletions.
Reviewed August 5, 2026 · model on record in the stance chip above.
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