{"id":"ec140dcb-04be-4496-a9b8-11e2d0efb199","arxiv_id":"1908.02385","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every rational number 1 + p/q with q > p^2 is shown to be the exact growth exponent of some bipartite Turan problem.","lead":"This paper proves that many fractions between 1 and 2, specifically all numbers of the form 1 + p/q with q > p^2, are realized as exact growth rates for bipartite Turan problems. It settles a large part of a long-standing conjecture of Erdos and Simonovits about rational Turan exponents.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma A.4's recursive construction in the proof of Lemma 3.2 does not exclude the spider S'_1 when choosing later spiders, so the asserted vertex-disjointness of the legs P_i is not established.","rationale":"The reader identified Lemma 3.2 as load-bearing and noted that it is only sketched. My stress-test agrees with that identification and locates a more specific unverified step inside the sketch: the recursive construction in Lemma A.4 does not explicitly force later spiders to avoid V(S'_1), even though the initial segments of the final legs live inside S'_1. Without such avoidance, the claimed vertex-disjointness of the legs P_i does not follow from the stated choices, and the construction may not produce a spider. This is a genuine gap in the written proof, but it appears repairable with a standard counting argument, so the verdict should remain conditional rather than being upgraded to reject or downgraded to accept. The strongest claim, Theorem 1.10, would collapse only if this gap cannot be repaired; the rest of the proof, especially Lemma 3.1, is detailed and appears sound.","tokens_in":23233,"tokens_out":22735,"duration_ms":205028,"concrete_test":"Write out the full recursion in Lemma A.4, adding V(S'_1)\\\\V(R_l) to the forbidden set at each step. Verify the counting: for each fixed forbidden vertex outside R_l, the number of extensions of R_l to a full spider containing that vertex is at most (K*delta)^(j - e(R_l) - 1), while |F|R_l| >= (K*delta)^(j - e(R_l))/L^2; the total forbidden set has size O(1), so for delta = omega(1) there remains a valid choice. If these counts work, the disjointness claim is settled; if not, exhibit a graph and family F satisfying Lemma A.3 where the original recursion yields intersecting P_i.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central upper bound, Theorem 1.10, rests on Lemma 3.2, which is proved only by a sketch in Appendix A. Lemma 3.2 is reduced to Lemma A.2, whose proof is Lemma A.4. In Lemma A.4, R0 is a sub-spider of some S'_1, and the initial segment of each final leg P_i is taken inside S'_1. The recursion then chooses S_{l+1} in F|R_l avoiding only Z and the previously chosen S_j,T_j (j <= l); it never forbids V(S'_1). Hence a later S_{l+1} or T_{l+1} can intersect the initial segments of P_i. The text asserts that P_1,...,P_s are vertex-disjoint and that T_{k+1} intersects union P_i only on its leaves, but this does not follow from the stated avoidance rules. If this disjointness fails, the final graph T = T_{k+1} union (union P_i) is not necessarily an s-legged spider, so Lemma A.2 may fail to produce t internally disjoint spiders. Since Lemma 3.2 uses that conclusion to contradict t*S^s_{b,k}-freeness, a failure here would invalidate Theorem 1.12 and hence Theorem 1.10. The gap appears repairable by adding V(S'_1) to the forbidden sets in all recursive choices and re-checking the size bounds, but as written the proof is incomplete.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23506,"tokens_out":16685,"duration_ms":169614,"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":[{"comment":"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":"Appendix A, Lemma A.4"},{"comment":"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.","section":"Section 3.1 and Appendix A, Lemma 3.2"}],"minor_comments":[{"comment":"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":"Section 3.2.2, Lemma 3.7"},{"comment":"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.","section":"Section 2, Lemma 2.4"},{"comment":"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.","section":"Proof of Theorem 1.12"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main result is likely correct and the new heavy-path arguments are a genuine contribution. The only serious concern is the completeness of Lemma 3.2: the appendix is a sketch and, more importantly, Lemma A.4 contains a concrete gap that currently breaks the proof. Since the gap appears readily repairable and the rest of the paper is meticulous, I recommend major revision rather than rejection. I would want to see a fully written proof of Lemma 3.2, or at least a precise statement of the Janzer lemma being extended and a complete proof of the extension, before accepting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know two things about arXiv:1908.02385. First, it proves a substantial new family of Turan exponents: every 1+p/q with q>p^2, via a three-parameter generalization of recent subdivision results. That is a real advance on the Erdős–Simonovits rational exponent conjecture, and the main new technique—Lemma 3.1, a bound on heavy paths in t*S^s_{b,k}-free graphs—is genuinely new and worked out in full detail. Second, the paper is not fully self-contained at a load-bearing point: Lemma 3.2, on heavy spiders, is only sketched in the appendix, and the sketch contains a specific gap.\n\nWhat is good: Theorem 1.10 covers all length vectors (b,k,...,k) with k≥b, extending earlier work of Janzer and of Conlon–Janzer–Lee that handled only b=k and b=1. The heavy-path lemma is the technical core, and the proof in Section 3.2 is careful, with a cleaning argument and an iterative spider build. The exposition is honest; the authors say exactly which parts are new and which are adapted.\n\nThe soft spot is in Lemma A.4. The construction of the final spider starts with a sub-spider R0 of some S'_1, and the initial segments of the legs P_i lie inside S'_1. When the recursion later chooses S_{ℓ+1} and T_{ℓ+1}, it avoids Z and the previously chosen S_j, T_j, but it never forbids V(S'_1). So a later spider can intersect those initial segments, and the asserted vertex-disjointness of the P_i does not follow. This is a real gap in the written proof. It looks repairable—add V(S'_1) to the forbidden set at every recursive step, and re-check the size bounds; S'_1 has at most sk vertices and L is large enough to absorb that. But as written, Lemma A.2, and hence Lemma 3.2 and Theorem 1.12, are not proven.\n\nI want to be clear: this is not a takedown. The main idea is sound, the gap is local, and the fix seems straightforward. But it is exactly the kind of thing a referee needs to see closed before the paper is published. The heavy-path lemma alone makes the paper worth reading.\n\nWho gains: extremal graph theorists, especially anyone working on Turan numbers of subdivisions or the rational exponent conjecture. I would take it to a reading group, and I would send it to a serious referee—asking for a complete proof of Lemma 3.2.\n\nBest,","headline":"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.","tokens_in":24093,"tokens_out":8402,"would_cite":true,"duration_ms":70716,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any positive integers p and q with q > p^2, the number 1 + p/q is a Turán exponent.","keywords":["Turán exponent","rational exponent conjecture","Turán number","subdivision","spider","heavy paths","extremal graph theory"],"falsifier":"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.","tokens_in":22995,"feed_emoji":"🕸️","tokens_out":9288,"duration_ms":95217,"temperature":0.7,"pith_summary":"The paper proves a new upper bound for Turán numbers of certain subdivisions of complete bipartite graphs, and from it derives that every rational number 1 + p/q with q > $p^{2}$ is a Turán exponent. A Turán exponent is a number r in (1,2) that occurs, up to a constant factor, as the maximum number of edges in a large graph avoiding some fixed bipartite subgraph. The result is a step toward the long-standing rational-exponent conjecture, which predicts every rational in (1,2) appears this way. The proof works by controlling paths and spiders that appear many times, and then assembling a forbidden subdivided complete bipartite graph.","feed_headline":"All rationals 1 + p/q with q > p^2 are Turán exponents","feed_subtitle":"A new upper bound for subdivided complete bipartite graphs realizes every rational exponent with denominator above p^2.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the lower bound for t-blowups of balanced rooted trees that matches the new upper bound and turns it into an exponent.","marker":"[2]"},{"why":"Introduces the admissible/light/heavy-path strategy and proves the b=1 case of the spider family.","marker":"[6]"},{"why":"Proves the b=k case for K^k_{s,t} and provides the lemma from which the sketched Lemma 3.2 is claimed to follow.","marker":"[16]"},{"why":"Extends light/heavy notions to spiders, handles earlier small cases, and supplies Lemma 2.5 used in the proof.","marker":"[20]"},{"why":"Provides the almost-regular reduction lemma used to reduce the Turán-number bound to the minimum-degree statement.","marker":"[21]"},{"why":"Gives the reduction lemma that turns the new realizable exponents into realizable values of the form 2−p/q.","marker":"[22]"}],"fun_headline_variants":["New Turán exponents for all q greater than p^2","Many rational Turán exponents via subdivisions","Erdős–Simonovits conjecture advances: q > p^2","Subdivisions unlock Turán exponents in new range"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["New Turán exponents for all q greater than p^2","Many rational Turán exponents via subdivisions","Erdős–Simonovits conjecture advances: q > p^2","Subdivisions unlock Turán exponents in new range"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000131,"raw_usage":{"total_tokens":1131,"prompt_tokens":948,"completion_tokens":183,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":117}},"tokens_in":564,"tokens_out":183,"duration_ms":2809,"temperature":1.0,"reasoning_tokens":117,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:45:19.636321+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Bukh and D","cited_arxiv_id":null,"evidence_quote":"Supplies the lower bound for t-blowups of balanced rooted trees that matches the new upper bound and turns it into an exponent."},{"cited_title":"More on the extremal number of subdivisions","cited_arxiv_id":"1903.10631","evidence_quote":"Introduces the admissible/light/heavy-path strategy and proves the b=1 case of the spider family."},{"cited_title":"The extremal number of the subdivisions of the complete bipartite graph","cited_arxiv_id":"1906.04084","evidence_quote":"Proves the b=k case for K^k_{s,t} and provides the lemma from which the sketched Lemma 3.2 is claimed to follow."},{"cited_title":"Turan numbers of bipartite subdivisions","cited_arxiv_id":"1905.08994","evidence_quote":"Extends light/heavy notions to spiders, handles earlier small cases, and supplies Lemma 2.5 used in the proof."},{"cited_title":"Jiang and R","cited_arxiv_id":null,"evidence_quote":"Provides the almost-regular reduction lemma used to reduce the Turán-number bound to the minimum-degree statement."},{"cited_title":"On the rational Tur\\'an exponents conjecture","cited_arxiv_id":"1811.06916","evidence_quote":"Gives the reduction lemma that turns the new realizable exponents into realizable values of the form 2−p/q."}],"review_version":1}