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Induced rational exponents and bipartite subgraphs in $K_{s, s}$-free graphs

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Pith's one-line read Every rational exponent in (1,2) is realized by an induced Turán family of size at most $2^a$ in $K_{s,s}$-free host graphs.

desk verdict Strong transfer results in induced Turán theory; one repairable counting error in Section 4.2. read the letter →

arxiv 2506.09020 v1 pith:LDLLEY6U submitted 2025-06-10 math.CO

Zichao Dong , Jun Gao , Ruonan Li , Hong Liu This is my paper
classification math.CO MSC 05C3505C38
keywords inducedTuránnumbersrationalexponentsK_{ss}-freegraphssupersaturationthetaprismtreesevencycles
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims a broad transfer phenomenon: many extremal results for bipartite graphs remain true when the forbidden condition is induced, provided the host graph is $K_{s,s}$-free. Concretely, for every rational $q=\frac{a}{b}\in(1,2)$, the paper constructs a family $\mathcal{H}$ of at most $2^a$ bipartite graphs whose induced Turán number is $\Theta_s(n^q)$, extending the Bukh–Conlon rational-exponent theorem to the induced setting. The engine is an optimal supersaturation result showing that dense, almost-regular $K_{s,s}$-free graphs contain many induced copies of every fixed tree, and analogous supersaturation for even cycles. Using the same machinery, the paper proves optimal-order induced Turán bounds for $\theta$ graphs and for even prism graphs, matching their known ordinary Turán exponents and giving strong evidence for the Hunter–Milojević–Sudakov–Tomon conjecture. A reader should care because the transfer makes a large body of bipartite extremal knowledge available to induced problems under a mild local-sparsity condition.

What carries the argument

The central objects are the $(p;S)$-lifts $F_p(T;R)$ of a rooted tree $(T;R)$: graphs obtained by gluing $p$ vertex-disjoint copies of $T$ along the root set $R$ and an additional set $S$, where $S$ ranges over subsets of the non-root vertices. The paper proves a Ramsey-type lemma that, from many induced copies of $T$ sharing a fixed root set, extracts $p$ copies whose intersections are regular and whose union is induced, forming a lift in $F_p(T;R)$. The extraction is powered by two ingredients: a greedy embedding that uses $K_{s,s}$-freeness to control bad common neighborhoods, and an almost-regular reduction (Lemma 2.1) that keeps degrees and codegrees bounded. For the cycle-based theorems, the machinery combines Sidorenko's property for even cycles, the Kővári–Sós–Turán theorem, and Janzer's regularization lemma to show that most closed $2\ell$-walks in a dense $K_{s,s}$-free graph are induced $2\ell$-cycles, from which induced $\theta$ and prism subgraphs are assembled.

What would settle it

Take the degenerate closed walk formed by two internally vertex-disjoint $\ell$-cycles meeting at exactly one vertex, and count the ordered pairs $(u,v)$ of vertices at distance $\ell$ along the walk: the walk lies in several distinct sets $A_{u,v}$, demonstrating that the inequality $\sum_{(u,v)\in\Gamma}|A_{u,v}|\le \#\{\text{degenerate $2\ell$-cycles}\}$ fails by a factor of at least $\ell$; checking whether the corrected factor still fits inside the slack constants of the proof settles whether Theorem 4.2 stands as written.

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Extended reading notes

Core claim

The central discovery is that the induced Turán number $\mathrm{ex}^*(n,\mathcal{H},s)$ — the maximum number of edges in an $n$-vertex $K_{s,s}$-free graph with no induced copy of any graph in $\mathcal{H}$ — satisfies the same rational exponents as the ordinary Turán number for a carefully chosen subfamily of the Bukh–Conlon lifting family. For every rational $q=\frac{a}{b}\in(1,2)$, the paper finds a family $\mathcal{H}$ of at most $2^a$ bipartite graphs with $\mathrm{ex}^*(n,\mathcal{H},s)=\Theta_s(n^q)$ (Theorem 1.2). It also proves that for $\theta$ graphs $\Theta^t_{\ell}$, $\mathrm{ex}^*(n,\Theta^t_{\ell},s)=\Theta_{\ell,t}(n^{1+1/\ell})$ up to $s$-dependent constants for all sufficiently large $t$ (Theorem 1.5), and that for even prism graphs $C^{\square}_{2\ell}$ with $\ell\ge 10$, $\mathrm{ex}^*(n,C^{\square}_{2\ell},s)=\Theta_{\ell}(n^{3/2})$ (Theorem 1.6). In both cases the $n$-exponent matches the ordinary Turán exponent, thereby confirming the transfer phenomenon for these families.

Load-bearing premise

The proof of Theorem 4.2 assumes that every degenerate homomorphic $2\ell$-cycle is counted in at most one of the sets $A_{u,v}$; a closed walk consisting of two cycles sharing a single vertex is counted in several such sets under different rotations, so the asserted uniqueness is false and the argument needs an extra factor of $\ell$.

Editorial extensions

If this is right

  • Every rational number in $(1,2)$ is the induced Turán exponent of a family of at most $2^a$ bipartite graphs, with the family size independent of $s$ and of $b$.
  • For theta graphs, the induced Turán number has the same $n$-exponent as the ordinary Turán number, $n^{1+1/\ell}$, for all sufficiently large $t$; for even prisms with $\ell\ge 10$, the induced Turán number is $\Theta_{\ell}(n^{3/2})$.
  • The supersaturation theorem shows that any $K$-almost-regular $K_{s,s}$-free graph with average degree $d$ contains at least $n(d/2K)^{t-1}$ labeled induced copies of every $t$-vertex tree, which is optimal up to constants.
  • The transfer phenomenon implies that ordinary bipartite Turán results can be lifted to induced statements in locally sparse hosts, making the Hunter–Milojević–Sudakov–Tomon conjecture plausible for a wider class of bipartite graphs.
  • The clique-blowup construction (Proposition 1.4) converts any ordinary lower bound $\mathrm{ex}(n,H)=\Omega(n^{1+\alpha})$ into an induced lower bound $\Omega_h(s^{1-\alpha}n^{1+\alpha})$, so for connected bipartite $H$ the induced Turán number is never much smaller than the ordinary one.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The double-counting gap in Section 4.2, where a degenerate closed walk formed by two cycles sharing a single vertex is counted in several sets $A_{u,v}$, is most likely absorbed by the very loose constants in the proof; a corrected factor of $\ell$ should not change the theorem's truth.
  • The transfer phenomenon probably extends to any bipartite $H$ whose ordinary Turán exponent is realized by lifted-tree constructions, as long as an induced-supersaturation statement holds for the building blocks; the paper's tree and even-cycle supersaturation results are natural templates.
  • The exponential $s$-dependence in the upper bounds for theta and prism graphs, contrasted with polynomial lower bounds, is an artifact of the method; closing this gap is a test of whether the transfer is tight in the parameters beyond $n$.
  • A further reduction from the $2^a$-element family to a single forbidden graph would settle Conjecture 1.1 but requires a fundamentally new idea, since the Ramsey cleaning step inherently needs many copies to find a regular intersection.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies induced Turán numbers under a K_{s,s}-freeness assumption on the host graph, proposing a transfer phenomenon from non-induced to induced extremal problems. The main results are: (i) every rational exponent q in (1,2) is realized as the induced Turán exponent of a subfamily of the Bukh–Conlon family of size at most 2^a (Theorem 1.2); (ii) supersaturation theorems for induced trees and induced even cycles in K_{s,s}-free almost-regular graphs (Theorems 1.3, 4.2, 5.2); and (iii) optimal-order upper bounds for induced theta graphs and induced even prisms (Theorems 1.5 and 1.6), matching the known non-induced exponents. The lower bounds are obtained by clique blowups of known constructions, while the upper bounds are proved by passing to almost-regular subgraphs and using auxiliary Ramsey and dependent-random-choice arguments.

Significance. If the proofs are correct, the paper gives substantial new evidence for the Hunter–Milojević–Sudakov–Tomon conjecture and for the proposed transfer phenomenon. The induced-rational-exponent result improves the Bukh–Conlon family size from an uncontrolled number to at most 2^a, and the induced theta and prism bounds match the best known non-induced exponents. The paper is largely self-contained: it supplies new supersaturation machinery, explicit constants, and a clean lower-bound construction via clique blowups. These are genuine contributions. The two technical gaps I identify below are localized and appear repairable within the paper's existing framework, so the high-level claims are plausible despite the need for revision.

major comments (2)
  1. [Section 4.2, Eq. (2)] The assertion before (2) that every degenerate homomorphic 2ℓ-cycle is counted in at most one set A_{u,v} is false. For ℓ=3, the closed walk a-b-c-a-d-c-a is degenerate, and it is counted in A_{b,d} when read starting at b with midpoint d and in A_{d,b} when read starting at d; the two ℓ-paths share the interior vertex c in both representations. More generally, a figure-eight walk can be counted under several rotations, so the left side of (2) should be bounded by O(ℓ) times the number of degenerate homomorphic cycles, not by that number itself. This error is load-bearing: with the uncorrected coefficient, the displayed inequality before the averaging step is no longer valid once the extra factor from (2) is inserted, and the derivation of the pair (u*,v*) is unsupported. The argument is repairable: replacing the coefficient of Σ|A| by its 1/(2ℓ) multiple (equivalently, enlarging the parameter δ in the application of Lemma 4.4 by a factor 2ℓ) is harmless because C>(ℓt)^{100ℓ} leaves enormous slack. As written, however, the proof of (2) and the subsequent averaging step are incorrect.
  2. [Section 3.2, proof of Theorem 1.3] The counting of embeddings does not exclude previously embedded vertices from the candidate set V_k. If w_i is an earlier leaf of the prefix tree whose only earlier neighbor is w_{k'}, then v_i lies in N(v_{k'}), is not in N(v_j) for any j≠k', and need not belong to any X(v_j); hence v_i is counted as an available choice for v_k even though reusing v_i would make the map non-injective. The lower bound |V_k|≥d/(2K) therefore overcounts valid embeddings. This gap is also repairable: subtracting at most t previously embedded vertices from each candidate set is absorbed by the slack in the assumption d≥(4Kt)^{6s}s^3, and the final constant in Theorem 1.3 would still have the required form. As stated, however, the proof does not justify the claimed number of induced copies.
minor comments (5)
  1. [Throughout] Displayed results are referred to by inconsistent labels: Lemma 2.1 is called Theorem 2.1 in its proof; Lemma 3.3 is cited as Theorem 3.3; Claim 3.4 as Theorem 3.4; Lemmas 4.3, 4.4, 5.3, 5.4, 5.5, 5.9 and Claim 5.10 are cited as Theorems 4.3, 4.4, 5.3, 5.4, 5.5, 5.9, 5.10; and Proposition 1.4 is called Theorem 1.4 in Sections 4 and 5. These labels should be reconciled.
  2. [Section 4.2, after Lemma 4.4] The text states that Lemma 4.4 yields 'at least δr²/2 = εr²/(64ℓ) pairs' among the selected paths, but the quantity used in Lemma 4.4 is the blue-edge parameter λ=ε/(32ℓ), not δ; the displayed equality is correct only if 'δ' is replaced by 'λ'.
  3. [Section 3.1, proof of Theorem 3.1] In the displayed lower bound for the number of induced copies of (T;R) under a fixed root orientation, the final term '≥ C/(4K)' appears to be missing the exponent and the cancellation of the powers of n; as written the algebra is unclear. The intended estimate should be that the ratio is at least (C/(4K))^{t-1}, since the exponent of n is 0 after substituting α=1-x/y and y=t-1.
  4. [Section 1.1 and Section 3.2] The notation for rational exponents is inconsistent: Theorem 1.2 writes q=a/b with q∈(1,2), while the remark in Section 3.2 discusses 'a/b∈(0,1)' as the deficit from 2 and uses rooted trees of density b/a. The change of variables should be stated explicitly to avoid confusion.
  5. [Section 5.3, Claim 5.6] The claim that at most 4 of the 2ℓ rotations of a homomorphic 2ℓ-cycle are special is asserted without proof; a one-sentence explanation of why the forbidden positions {1,ℓ+1} account for at most four rotations per edge would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; upper bounds derived internally, lower bounds from external constructions; only a minor non-load-bearing self-citation to the authors' prism paper.

full rationale

This paper derives its main upper bounds (Theorems 1.2, 1.5, 1.6) from first-principles supersaturation results (Theorems 1.3, 4.2, 5.2) proved inside the paper using KST-type lemmas and external tools (Sidorenko, Janzer, Hunter et al. lemmas). Lower bounds are imported from external constructions (Bukh-Conlon rational exponent families, Conlon's theta graphs, known C4 lower bounds) via Proposition 1.4. The only self-citation is [23] (Gao-Janzer-Liu-Xu), which includes an author; it is used merely as a proof-strategy precedent in Section 5.1 ('We use a similar strategy as in the proof of [23, Theorem 1.1]'), and the induced-prism proof is fully self-contained, so the citation is not load-bearing. I find no step where a claimed prediction reduces by definition to a fitted parameter or to a self-citation chain. The skeptic's concern about the assertion in Section 4.2 that each degenerate 2-ell cycle is counted in at most one A_{u,v} is a potential correctness issue (the uniqueness claim appears unproved and may be false under multiple rotations), but an overcount factor is not a circular reduction of the conclusion to the hypothesis; it is a bounded-error artifact repairable by a constant factor. No circularity is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper imports standard external theorems for lower bounds and technical lemmas, and its own new proof components are self-contained derivations. No free parameters are fitted to empirical data, and no new mathematical entities are postulated. The central claim therefore carries a low circularity burden.

assumptions (7)
  • standard math Kővári-Sós-Turán theorem: ex(n,K_{s,s}) is O(n^{2-1/s})
    Used in Corollary 2.3 and throughout Sections 3-5 to bound edge counts in K_{s,s}-free graphs.
  • standard math Turán's theorem for graphs with bounded independence number
    Used in Section 3.1 to lower-bound edges in the auxiliary graph H on induced copies.
  • standard math Sidorenko property for even cycles: hom(C_{2ℓ},G) is at least d^{2ℓ}
    Used in Theorem 4.2 and Section 5.3 to get many closed walks in dense graphs.
  • domain assumption Bukh-Conlon random algebraic construction supplies ex(n,T)=Ω(n^q) for their rooted-tree families
    External lower bound imported from [7]; it sets the lower bound for Theorem 1.2.
  • domain assumption Conlon's theorem ex(n,Θ^t_ℓ)=Θ(n^{1+1/ℓ}) for t sufficiently large
    External result [10] gives the lower bound in Theorem 1.5.
  • standard math External lemmas from Hunter et al. [28] and Janzer [30] used for degenerate walks, dependent random choice, and regularization are correct
    The paper quotes Lemmas 4.3, 4.4, 5.3, 5.9, 5.4 and 5.5 from published work without proof; correctness is assumed.
  • standard math ex(n,C4)=Ω(n^{3/2})
    Classical lower bound used for the prism lower bound in Theorem 1.6.

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Pith. "Pith review of Induced rational exponents and bipartite subgraphs in $K_{s, s}$-free graphs." pith.science (2026). https://pith.science/paper/LDLLEY6U

@misc{pith2026250609020,
  author       = {Pith},
  title        = {Pith review of: Induced rational exponents and bipartite subgraphs in $K_s, s$-free graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LDLLEY6U}},
  note         = {Machine review of arXiv:2506.09020}
}
abstract

In this paper, we study a general phenomenon that many extremal results for bipartite graphs can be transferred to the induced setting when the host graph is $K_{s, s}$-free. As manifestations of this phenomenon, we prove that every rational $\frac{a}{b} \in (1, 2), \, a, b \in \mathbb{N}_+$, can be achieved as Tur\'{a}n exponent of a family of at most $2^a$ induced forbidden bipartite graphs, extending a result of Bukh and Conlon [JEMS 2018]. Our forbidden family is a subfamily of theirs which is substantially smaller. A key ingredient, which is yet another instance of this phenomenon, is supersaturation results for induced trees and cycles in $K_{s, s}$-free graphs. We also provide new evidence to a recent conjecture of Hunter, Milojevi\'{c}, Sudakov, and Tomon [JCTB 2025] by proving optimal bounds for the maximum size of $K_{s, s}$-free graphs without an induced copy of theta graphs or prism graphs, whose Tur\'{a}n exponents were determined by Conlon [BLMS 2019] and by Gao, Janzer, Liu, and Xu [IJM 2025+].

Figures

Figures reproduced from arXiv: 2506.09020 by the authors.

Figure 1
Figure 1. The prism graph C □ 10. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The (p; S)-lift of a rooted 4-edge path where R = {a, e}, p = 3, and S = {d}. To conclude Theorem 1.2 using Theorem 1.4, the following upper bound suffices. Theorem 3.1. Let t ≥ 2, s ≥ 2 be integers and suppose (T; R) is a t-vertex rooted tree. There exists C = C(s, t, p) > 0 such that every n-vertex Ks,s-free graph G with e(G) ≥ C · n 2− 1 ρ(T ;R) contains an induced F-subgraph for some F ∈ Fp (T; R) (i.e., F is a … view at source ↗
Figure 3
Figure 3. An illustration of a graph living in G 3 \ F3 . Remark 3.2. It is worth mentioning that our F p (T; R) is smaller than the Bukh–Conlon family G p (T; R) (see [7, Definition 1.2], denoted as T p R ), which they refer to as the p-th power of a fixed rooted tree (T; R). The family G p (T; R) consists of all possible unions of p distinct labeled copies of T which agree on R. Omit (T; R) when there is no confusion. For t… view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Supersaturation of induced even cycles in locally sparse graphs

    math.CO 2026-08 accept novelty 7.0 of 10

    For every integer ℓ≥2, any (1−ε,t)-sparse n-vertex graph with at least C t^{1−1/ℓ} n^{1+1/ℓ} edges contains at least C' d^{2ℓ} induced copies of the even cycle C_{2ℓ}, where d is its average degree.

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Works this paper leans on

45 extracted references · 38 canonical work pages · cited by 1 Pith paper

  1. [1]

    N. Alon, M. Krivelevich, and B. Sudakov. Tur´ an numbers of bipartite graphs and related Ramsey- type questions.Combin. Probab. Comput., 12(5-6):477–494, 2003. Special issue on Ramsey theory

    work page 2003

  2. [2]

    C. T. Benson. Minimal regular graphs of girths eight and twelve.Canadian J. Math., 18:1091–1094, 1966

    work page 1966

  3. [3]

    J. A. Bondy and M. Simonovits. Cycles of even length in graphs.J. Combin. Theory Ser. B, 16:97–105, 1974

    1974

  4. [4]

    Bourneuf, M

    R. Bourneuf, M. Buci´ c, L. Cook, and J. Davies. On polynomial degree-boundedness.Adv. Comb., pages Paper No. 5, 16, 2024

    work page 2024

  5. [5]

    W. G. Brown. On graphs that do not contain a Thomsen graph.Canad. Math. Bull., 9:281–285, 1966

    work page 1966

  6. [6]

    B. Bukh. Random algebraic construction of extremal graphs.Bull. Lond. Math. Soc., 47(6):939– 945, 2015

    2015

  7. [7]

    Bukh and D

    B. Bukh and D. Conlon. Rational exponents in extremal graph theory.J. Eur. Math. Soc. (JEMS), 20(7):1747–1757, 2018

    work page 2018

  8. [8]

    Bukh and Z

    B. Bukh and Z. Jiang. A bound on the number of edges in graphs without an even cycle.Combin. Probab. Comput., 26(1):1–15, 2017

    work page 2017

Show all 45 references
  1. [9]

    Bukh and M

    B. Bukh and M. Tait. Tur´ an numbers of theta graphs.Combin. Probab. Comput., 29(4):495–507, 2020. 19

    2020

  2. [10]

    D. Conlon. Graphs with few paths of prescribed length between any two vertices.Bull. Lond. Math. Soc., 51(6):1015–1021, 2019

    2019

  3. [11]

    Conlon and O

    D. Conlon and O. Janzer. Rational exponents near two.Adv. Comb., pages Paper No. 9, 10, 2022

    2022

  4. [12]

    Conlon, O

    D. Conlon, O. Janzer, and J. Lee. More on the extremal number of subdivisions.Combinatorica, 41(4):465–494, 2021

    2021

  5. [13]

    Corsten and T

    J. Corsten and T. Tran. Balanced supersaturation for some degenerate hypergraphs.J. Graph Theory, 97(4):600–623, 2021

    2021

  6. [14]

    Dubroff, B

    Q. Dubroff, B. Gunby, B. Narayanan, and S. Spiro. Clique Supersaturation, 2023. arXiv:2312.08265

    2023 arXiv

  7. [15]

    P. Erd˝ os. Problems and results in graph theory. InThe theory and applications of graphs (Kala- mazoo, Mich., 1980), pages 331–341. Wiley, New York, 1981

    1980

  8. [16]

    Erd˝ os and M

    P. Erd˝ os and M. Simonovits. A limit theorem in graph theory.Studia Sci. Math. Hungar., 1:51–57, 1966

    1966

  9. [17]

    Erd˝ os and A

    P. Erd˝ os and A. H. Stone. On the structure of linear graphs.Bull. Amer. Math. Soc., 52:1087–1091, 1946

    1946

  10. [18]

    P. Erd˝ os. Extremal problems in graph theory. InTheory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), pages 29–36. Publ. House Czech. Acad. Sci., Prague, 1964

    1963

  11. [19]

    Erd˝ os and M

    P. Erd˝ os and M. Simonovits. Some extremal problems in graph theory. InCombinatorial theory and its applications, I-III (Proc. Colloq., Balatonf¨ ured, 1969), volume 4 ofColloq. Math. Soc. J´ anos Bolyai, pages 377–390. North-Holland, Amsterdam-London, 1970

    1969

  12. [20]

    R. J. Faudree and M. Simonovits. On a class of degenerate extremal graph problems.Combina- torica, 3(1):83–93, 1983

    1983

  13. [21]

    F¨ uredi

    Z. F¨ uredi. On a Tur´ an type problem of Erd˝ os.Combinatorica, 11(1):75–79, 1991

    1991

  14. [22]

    F¨ uredi and M

    Z. F¨ uredi and M. Simonovits. The history of degenerate (bipartite) extremal graph problems. In Erd¨ os centennial, volume 25 ofBolyai Soc. Math. Stud., pages 169–264. J´ anos Bolyai Math. Soc., Budapest, 2013

    2013

  15. [23]

    J. Gao, O. Janzer, H. Liu, and Z. Xu. Extremal number of graphs from geometric shapes.Israel J. of Math., to appear

  16. [24]

    Gir˜ ao and Z

    A. Gir˜ ao and Z. Hunter. Induced subdivisions inKs,s-free graphs with polynomial average degree. Int. Math. Res. Not. IMRN, 4:Paper No. rnaf025, 23, 2025

    2025

  17. [25]

    X. He, Y. Li, and L. Feng. Extremal graphs for the odd prism.Discrete Math., 348(1):Paper No. 114249, 17, 2025

    2025

  18. [26]

    Z. He. A new upper bound on the Tur´ an number of even cycles.Electron. J. Combin., 28(2):Paper No. 2.41, 18, 2021

    2021

  19. [27]

    He and M

    Z. He and M. Tait. Hypergraphs with few Berge paths of fixed length between vertices.SIAM J. Discrete Math., 33(3):1472–1481, 2019

    2019

  20. [28]

    Hunter, A

    Z. Hunter, A. Milojevi´ c, B. Sudakov, and I. Tomon. K˝ ov´ ari-S´ os-Tur´ an theorem for hereditary families.J. Combin. Theory Ser. B, 172:168–197, 2025. 20

    2025

  21. [29]

    O. Janzer. The extremal number of the subdivisions of the complete bipartite graph.SIAM J. Discrete Math., 34(1):241–250, 2020

    2020

  22. [30]

    O. Janzer. Disproof of a conjecture of Erd˝ os and Simonovits on the Tur´ an number of graphs with minimum degree 3.Int. Math. Res. Not. IMRN, 10:8478–8494, 2023

    2023

  23. [31]

    Jiang, Z

    T. Jiang, Z. Jiang, and J. Ma. Negligible obstructions and Tur´ an exponents.Ann. Appl. Math., 38(3):356–384, 2022

    2022

  24. [32]

    D. Kang, J. Kim, and H. Liu. On the rational Tur´ an exponents conjecture.J. Combin. Theory Ser. B, 148:149–172, 2021

    2021

  25. [33]

    K¨ ovari, V

    T. K¨ ovari, V. T. S´ os, and P. Tur´ an. On a problem of K. Zarankiewicz.Colloq. Math., 3:50–57, 1954

    1954

  26. [34]

    Lazebnik, V

    F. Lazebnik, V. A. Ustimenko, and A. J. Woldar. Properties of certain families of 2k-cycle-free graphs.J. Combin. Theory Ser. B, 60(2):293–298, 1994

    1994

  27. [35]

    J. Ma, X. Yuan, and M. Zhang. Some extremal results on complete degenerate hypergraphs.J. Combin. Theory Ser. A, 154:598–609, 2018

    2018

  28. [36]

    W. Mantel. Problem 28.Wiskundige Opgaven, 10:60–61, 1907

    1907

  29. [37]

    Pikhurko

    O. Pikhurko. A note on the Tur´ an function of even cycles.Proc. Amer. Math. Soc., 140(11):3687– 3692, 2012

    2012

  30. [38]

    Pikhurko and Z.B

    O. Pikhurko and Z.B. Yilma. Supersaturation problem for color-critical graphs.J. Combin. Theory Ser. B, 123:148–185, 2017

    2017

  31. [39]

    Pinchasi and M

    R. Pinchasi and M. Sharir. On graphs that do not contain the cube and related problems.Com- binatorica, 25(5):615–623, 2005

    2005

  32. [40]

    Scott, P

    A. Scott, P. Seymour, and S. Spirkl. Polynomial bounds for chromatic number. I. Excluding a biclique and an induced tree.J. Graph Theory, 102(3):458–471, 2023

    2023

  33. [41]

    Sidorenko

    A. Sidorenko. Inequalities for functionals generated by bipartite graphs.Diskret. Mat., 3(3):50–65, 1991

    1991

  34. [42]

    P. Tur´ an. Eine Extremalaufgabe aus der Graphentheorie.Mat. Fiz. Lapok, 48:436–452, 1941

    1941

  35. [43]

    Verstra¨ ete and J

    J. Verstra¨ ete and J. Williford. Graphs without theta subgraphs.J. Combin. Theory Ser. B, 134:76–87, 2019

    2019

  36. [44]

    R. Wenger. Extremal graphs with noC 4’s,C 6’s, orC 10’s.J. Combin. Theory Ser. B, 52(1):113– 116, 1991

    1991

  37. [45]

    Z. Xu, T. Zhang, and G. Ge. Some tight lower bounds for Tur´ an problems via constructions of multi-hypergraphs.European J. Combin., 89:103161, 11, 2020. 21

    2020

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