REVIEW 2 major objections 4 minor 27 references
Many Turan exponents via subdivisions
T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read For any positive integers p and q with q > p^2, the number 1 + p/q is a Turán exponent.
desk verdict A major step on the rational exponent conjecture with a genuinely new heavy-path argument, but the heavy-spider lemma's appendix proof has a real gap that looks repairable. 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 argument is carried by a recursive notion of j-admissible, j-light, and j-heavy paths and spiders. A path or spider is admissible when all its proper subobjects are light; it is light when fewer than f(j,L) admissible objects share its endpoints or leaf vector, and heavy otherwise, where f(j,L) is a rapidly growing threshold chosen so that f(j,L) dominates all earlier thresholds. The key lemmas show that in a t*S^s_{b,k}-free almost-regular graph there are very few heavy paths, and few heavy spiders with arbitrary leg lengths; then a cleaning argument extracts a large family of light spiders whose size contradicts the light threshold. The genuinely new part is handling heavy paths of length at most (k+b)/2, which requires building a well-placed spider of height k from light-path extensions and then merging it with many internally disjoint paths to obtain the forbidden blowup.
What would settle it
Check Lemma 3.2 for a parameter case not covered by earlier work, say s=3, b=2, k=4 with an allowed leg-length vector: if the number of heavy spiders exceeds $27K^{{j−2}}$$L^{{-1}}$ n δ^j for some vector, the claimed bound is false and Theorem 1.12 fails.
Extended reading notes
Core claim
For fixed s,t ≥ 2 and k ≥ b ≥ 1, let S^s_{b,k} be the s-legged spider with one leg of length b and the other s−1 legs of length k, and let t*S^s_{b,k} denote the union of t copies sharing the leaves. The paper establishes ex(n, t*S^s_{b,k}) = O($n^{{1+(s−1)/((s−1)k+b)}}$), matching a known lower bound and generalizing earlier results that covered b=k and b=1. Consequently, the three-parameter family 1+p/(kp+b), with k ≥ b ≥ 1, consists of Turán exponents. Since every fraction p/q with q > $p^{2}$ can be written as p/(kp+b), this implies 1+p/q is a Turán exponent for all q > $p^{2}$. The paper also derives complementary exponents of the form 2 − (kp+b)/(s(kp+b)+p) and, in particular, 2 − p/q whenever q > p and q mod p ≤ √p.
Load-bearing premise
Lemma 3.2, which bounds heavy spiders with arbitrary leg lengths, is only sketched and is said to extend a lemma from another paper; if that extension is not valid, the proof of the main upper bound collapses.
Editorial extensions
If this is right
- For every pair of positive integers p,q with q>p^2, the graph constructed here shows the exponent 1+p/q is realized by a bipartite graph.
- All exponents 1+p/(kp+b) with k≥b≥1 are realized by t-blowups of one-spider subdivisions, subsuming the previously known cases b=k and b=1.
- Via the reduction lemma from [22], 2−(kp+b)/(s(kp+b)+p) is also a Turán exponent for k≥b−1, so new exponents near 2 follow as well.
- The result establishes all the rational exponents promised by the general spider conjecture, without resolving that conjecture's full upper bound.
Reading between the lines
- My inference: the short-heavy-path method may extend to spiders with more than one shortened leg; if it does, the full spider conjecture would follow and would realize exponents with denominators closer to p.
- My inference: the same cleaning-by-threshold technique could be tried on balanced rooted trees that are not spiders, where matching upper bounds are still open.
- My inference: a testable consequence is that for any fixed p, the smallest denominator q for which this one-short-leg construction fails should grow quadratically in p; replacing the single short leg by a different length pattern may lower that threshold.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies Turán numbers of subdivisions of complete bipartite graphs with varying subdivision lengths. Its main result (Theorem 1.10) is that for any integers s, t ≥ 2 and k ≥ b ≥ 1, the t-blowup of the s-legged spider with length vector (b, k, ..., k) satisfies ex(n, t*S^s_{b,k}) = O(n^{1+(s-1)/((s-1)k+b)}). Combined with the Bukh–Conlon lower bound, this yields that 1 + p/(kp+b) is a Turán exponent for all positive integers p, k, b with k ≥ b, and in particular that every rational of the form 1 + p/q with q > p^2 is a Turán exponent. The proof adapts the framework of admissible, light, and heavy paths and spiders developed by Conlon–Lee, Conlon–Janzer–Lee, Jiang–Qiu, and Janzer. The main new contribution is a heavy-path lemma (Lemma 3.1), proved in full in Sections 3.2–3.2.3, which controls the number of heavy paths of every length in a t*S^s_{b,k}-free almost-regular graph. The companion lemma on heavy spiders (Lemma 3.2) is deferred to Appendix A, where only a sketch following Janzer is provided.
Significance. If the proof is correct, the paper makes substantial progress on the Erdős–Simonovits rational exponent conjecture: it establishes all rationals 1 + p/q with q > p^2, a large family not previously known. The result unifies and extends recent theorems of Conlon–Janzer–Lee and Janzer, and the proofs of Lemmas 3.1, 3.5, 3.7 and 3.8 contain genuinely new ideas that are likely to be useful for further attacks on Conjectures 1.7 and 1.9. The exponents are not fitted: the lower bound comes from the cited Bukh–Conlon theorem and the upper bound is proved directly from graph-theoretic definitions, with the constants chosen to satisfy inequalities rather than to match a target exponent. The paper is clearly written, and the main new lemma (Lemma 3.1) is proved in full detail with explicit constants.
major comments (2)
- [Appendix A, Lemma A.4] The recursive construction in Lemma A.4 does not exclude the vertices of the initially chosen spider S'_1 (or more precisely the segments of S'_1 that will become the initial parts of the legs P_i) from later choices of S_{ℓ+1} and T_{ℓ+1}. The choices for S_{ℓ+1} and T_{ℓ+1} only avoid Z, the previously chosen S_j's, and the previously chosen T_j's; V(S'_1) is never added to the forbidden set. Consequently, a later S_{ℓ+1} or T_{ℓ+1} can intersect the path from v_i to x_{1,i} inside S'_1. The assertions that P_1,...,P_s are vertex-disjoint and that T_{k+1} meets their union only in its leaves do not follow from the stated avoidance rules. If these fail, the constructed graph T is not necessarily an s-legged spider with the claimed leaf vector, so Lemma A.2 may fail to produce t internally disjoint spiders. This matters because Lemma 3.2 uses exactly that conclusion to contradict t*S^s_{b,k}-freeness. The gap appears repairable by adding V(S'_1) (or the relevant initial segments) to the forbidden set in all recursive choices, and the size bounds in the proof seem to allow this, but as written the proof is incomplete.
- [Section 3.1 and Appendix A, Lemma 3.2] Lemma 3.2 is load-bearing: it is used in the proof of Theorem 1.12 to bound the number of heavy spiders with an arbitrary length vector (j_1,...,j_s), and Theorem 1.12 is the technical core of the main upper bound. However, the lemma is not proved in the body of the paper; the appendix is introduced as a 'sketch' and relies on an unproved extension of Janzer's Lemma 4.3 in [16]. The manuscript should either provide a complete proof of Lemma 3.2 in the stated generality, or state the precise lemma from [16] and prove that this extension is valid. A sketch is not sufficient for a lemma on which the central claim depends.
minor comments (4)
- [Section 3.2.2, Lemma 3.7] In the statement of Lemma 3.7, part 1, the leaf set of T is said to be contained in A_b and that of T' in A_{b-1}; however, b may be larger than j, while A_i is defined only for 0 ≤ i ≤ j. The proof shows the intended sets are A_{b'} and A_{b'-1} (and similarly for the odd case, A_{j-b'} and A_{j-b'+1}). Please correct the statement.
- [Section 2, Lemma 2.4] In the first line of the proof of Lemma 2.4, 'Let S′ = {S_1,...,S_r} ⊆ C' refers to a set C that has not been defined; it should be the given family S. Also, the notation S′ conflicts with the name of the original family; a different letter (e.g., M) would improve readability.
- [Proof of Theorem 1.12] The lower bound |S| ≥ (1-o(1))/(h+1)! · n δ^h is asserted with 'a greedy process' without further detail. A brief explanation (counting choices of each leg sequentially) would help the reader verify the constant.
- [Throughout] There are several typographical errors, e.g., 'K-almost-egular' in Lemma 2.5 and 'Corollay 1.4' in Section 4. These do not affect the mathematics but should be corrected in a revision.
Circularity Check
No circularity: the new Turan exponents are obtained from an external lower bound and a directly proved upper bound, with constants chosen to satisfy inequalities rather than to match the target exponent.
full rationale
The paper's central claim, Theorem 1.2, is derived by combining the Bukh-Conlon lower bound (Theorem 1.6, an external result) with the paper's own upper bound (Theorem 1.10). The upper bound is proved by a direct graph-theoretic argument: assuming many heavy paths or heavy spiders in a K-almost-regular t*S^s_{b,k}-free graph, the proof constructs a forbidden copy of t*S^s_{b,k}. The exponent (s-1)/((s-1)k+b) is fixed by the definition of the host graph and appears in the assumed minimum-degree threshold that is contradicted; it is not fitted. The auxiliary function f(j,L) and the constants L, C, M, D are chosen recursively to make counting inequalities work, not to reverse-engineer the final exponent. The lower-bound side is taken verbatim from Bukh and Conlon's theorem, and the upper-bound side does not assume the conclusion. The paper does invoke Janzer's prior work for Lemma 3.2, but this is an external source, not a self-citation, and the appendix provides a sketch intended to make the paper self-contained. The skeptical concern about Lemma A.4 is a potential gap in the proof of vertex-disjointness of the constructed legs, not a circularity: even if the sketch is incomplete, the argument does not assume the target Turan exponent or redefine the problem in terms of its conclusion. The paper is not self-referential in a load-bearing way, and no prediction or derived quantity is equivalent to an input by construction. Therefore the circularity score is 0.
Assumptions & free parameters
assumptions (4)
- domain assumption Theorem 1.6 (Bukh-Conlon lower bound): for every balanced rooted tree, ex(n, t*T_R) = Omega(n^(2-1/rho)).
- domain assumption Lemma 1.11 (Erdos-Simonovits regularization, as formulated in Jiang-Seiver): every dense graph contains a K-almost-regular subgraph.
- domain assumption Lemma 2.5 (spider extension lemma from Jiang and Qiu [20]).
- domain assumption Lemma 4.1 (Kang-Kim-Liu reduction): if 2 - a/b is balancedly realizable, then 2 - a/(a+b) is too.
Cite this review
Pith. "Pith review of Many Turan exponents via subdivisions." pith.science (2026). https://pith.science/paper/GQOGR3HQ
@misc{pith2026190802385,
author = {Pith},
title = {Pith review of: Many Turan exponents via subdivisions},
year = {2026},
howpublished = {\url{https://pith.science/paper/GQOGR3HQ}},
note = {Machine review of arXiv:1908.02385}
}
abstract
Given a graph $H$ and a positive integer $n$, the {\it Tur\'an number} $\ex(n,H)$ is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. A real number $r\in(1,2)$ is called a {\it Tur\'an exponent} if there exists a bipartite graph $H$ such that $\ex(n,H)=\Theta(n^r)$. A long-standing conjecture of Erd\H{o}s and Simonovits states that $1+\frac{p}{q}$ is a Tur\'an exponent for all positive integers $p$ and $q$ with $q> p$. In this paper, we build on recent developments on the conjecture to establish a large family of new Tur\'an exponents. In particular, it follows from our main result that $1+\frac{p}{q}$ is a Tur\'an exponent for all positive integers $p$ and $q$ with $q> p^2$.
Reference graph
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For each S ∈ F , at least f (j, L)/2 member of F share the same leaf vector as S
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For any T ∈ ∂(S), |F |T | ≥ (Kδ )j−e(T )/L2. Proof. Let F ∗ be the family of all heavy spiders in G with length vector ( j1, . . . , js). Suppose that |F ∗| ≥ nδj L . For each vector ( x1, . . . , xs) of s distinct vertices in G, let F ∗ (x1,...,xs) denote the subfamily of mem...
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is a family of internally disjoint spiders of size L2 > |Z| + 2, there exist members S1, T1 of T (S′
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Let R1 be the subspider of T1 with length vector (j1 − η1,1,
such that S1 and T1 are disjoint from Z. Let R1 be the subspider of T1 with length vector (j1 − η1,1, . . . , js − ηs,1). 19 Iteratively, for 1 ≤ ℓ ≤ k, suppose we have defined Rℓ of length vector ( j1−η1,ℓ, . . . , js − ηs,ℓ) which is a subspider of Tℓ ∈ F . We define Sℓ+1, Tℓ+...
Reviewed August 14, 2026 · model on record in the stance chip above.
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