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Survey of generalized Tur\'an problems -- counting subgraphs

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Pith's one-line read A survey of generalized Turán problems claims that counting copies of a subgraph H in F-free graphs is now a mature field with a known map, a toolkit of eight methods, and a concrete open-problem list.

desk verdict A genuinely useful survey that is let down by one incorrect proof sketch; the rest holds up. read the letter →

arxiv 2506.03418 v1 pith:P75FB3WW submitted 2025-06-03 math.CO

Dániel Gerbner , Cory Palmer This is my paper
classification math.CO MSC 05C3505C3005D40
keywords generalizedTuránnumbersextremalgraphtheorysubgraphcountingF-freegraphsTurán-goodTurán-stableflagalgebrasdegenerateproblems
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

The paper surveys the generalized Turán number $\mathrm{ex}(n,H,F)$, the maximum number of copies of a subgraph $H$ in an $n$-vertex graph containing no copy of $F$. It aims to give an exhaustive inventory of what is known about this function, arguing that the field has a clear structure: a non-degenerate case governed by chromatic numbers and 'Turán-good'/'Turán-stable' graphs, and a degenerate case organized by the graph $H$ being counted. The survey traces the subject from pre-2016 results such as Zykov's 1949 theorem and Erdős's 1938 cherry bound through Alon and Shikhelman's systematic program and the subsequent boom, then catalogs eight methods and a list of open problems. A sympathetic reader would take the paper's main contribution to be the map itself: a reliable, organized reference for what is known, what methods prove it, and where the gaps are.

What carries the argument

The central object is the function $\mathrm{ex}(n,H,F)$, together with the dichotomy supplied by the blowup criterion: $F$ embeds in a blowup of $H$ if and only if $\mathrm{ex}(n,H,F)=o(n^{|V(H)|})$. The named structural properties are $F$-Turán-good and $F$-Turán-stable (plus weak variants), which say respectively that the Turán graph, or some complete multipartite graph, maximizes copies of $H$ asymptotically or exactly; the survey uses stability of these properties as a machine for converting approximate structural statements into exact formulas. The method section adds the standard tools—regularity lemma, probabilistic method, stability method, flag algebras, spectral bounds, progressive induction, and algebraic constructions—as the practical machinery behind the inventory.

What would settle it

Test the dropped condition with $H=C_5$ and $F=K_3$: look for $n$-vertex graphs with $\varepsilon n^5$ copies of $C_5$ but only $o(n^3)$ triangles; their existence would refute Theorem 2.8 as printed.

Watch

Extended reading notes

Core claim

The paper's central claim, on its own terms, is that generalized Turán numbers $\mathrm{ex}(n,H,F)$ now form a coherent and fully surveyed field, and that this survey provides the map. The organizing classification is Proposition 2.3: $\mathrm{ex}(n,H,F)=\Theta(n^{|V(H)|})$ exactly when $F$ is not a subgraph of a blowup of $H$, and otherwise the function is $o(n^{|V(H)|})$. In the non-degenerate case with $\chi(H)<\chi(F)$, the paper highlights Turán-good and Turán-stable graphs as the properties that yield exact extremal graphs, including an extension of the Simonovits color-critical-edge theorem to counting cliques. In the degenerate case, the survey groups results by the counted graph $H$—triangles, cliques, complete multipartite graphs, cycles, stars, forests, and other graphs—and records the best-known bounds, many of which remain open. The paper also devotes a section to eight methods and closes with a list of open problems.

Load-bearing premise

The load-bearing premise is the survey's unsupported claim that the technical condition in the supersaturation theorem of Halfpap and Palmer—the counted graph having smaller chromatic number than the forbidden graph—can be dropped without changing the conclusion, so if that assertion fails, Theorem 2.8 as printed fails.

Editorial extensions

If this is right

  • If the inventory is accurate, anyone working on $\mathrm{ex}(n,H,F)$ can first locate the pair $(H,F)$ in the non-degenerate or degenerate section and find the best known bound and the right method.
  • The stability-to-exact pipeline shows that proving $H$ is $k$-Turán-stable automatically yields exact results for $\mathrm{ex}(n,H,F)$ for every $k$-chromatic $F$ with a color-critical edge, so one stability proof produces infinitely many exact theorems.
  • Several open problems in the degenerate case are explicitly tied to classical Turán conjectures, so progress on, say, the Erdős girth conjecture or the Erdős–Sós conjecture would immediately sharpen generalized Turán bounds.
  • The open-problem list singles out concrete targets, including exact values of $\mathrm{ex}(n,C_6,C_8)$ and $\mathrm{ex}(n,P_6,P_8)$, and Conjecture 6.3 on cliques in tree-free graphs.

Reading between the lines

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

  • Beyond the paper: the survey's completeness claim is the kind of assertion that a citation-graph analysis could test, since the authors admit that results hidden inside other proofs may have been missed and the map would then need periodic revision.
  • Beyond the paper: the unproved strengthening of the supersaturation theorem is a natural target for a small computational search over graphs $H$ and $F$ with $\chi(H)\ge\chi(F)$, since a counterexample would change Theorem 2.8 as printed.
  • Beyond the paper: the star-counting results connected to degree-power indices suggest that invariants studied in chemical graph theory, such as Zagreb and Randić indices, could be a source of new generalized Turán bounds and vice versa.
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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 / 4 minor

Summary. The paper is a survey of the generalized Turán number ex(n,H,F), the maximum number of copies of a fixed graph H in an n-vertex F-free graph. It covers the history of the subject, general theorems (Alon–Shikhelman, Gerbner–Palmer, supersaturation and stability), the non-degenerate and degenerate cases organized by the graph H, a methods section (regularity, probabilistic, stability, flag algebras, spectral, progressive induction, algebraic constructions), and a list of open problems. The authors state an aim of exhaustively collecting the literature and introducing unifying terminology (k-Turán-good, F-Turán-stable, t-Füredi-good).

Significance. If the survey is accurate, it provides a valuable and much-needed central reference for a rapidly growing area. Its systematic terminology and organization of scattered results, together with proof sketches of representative methods, make it useful both to specialists and to newcomers. The discussion of methods and open problems is a genuine contribution. However, the survey's value depends on the reliability of its statements and proofs; the proof of the probabilistic lower bound in Proposition 5.7 is incorrect as printed, and the survey incorporates results from two 'in preparation' works, which weakens the claimed exhaustiveness and verifiability.

major comments (2)
  1. [Section 5.2, Proposition 5.7 (also Proposition 2.4)] The proof of Proposition 5.7 is mathematically invalid as written. With the stated constant c = h^h(e(F)-e(H)) + 1, the expected number of copies of H destroyed by deleting one edge per copy of F is larger than the expected number of copies of H in the random graph, so the difference displayed in the proof is negative. Explicitly, the H-count term has coefficient c^{e(H)}/h^h and the deletion term has coefficient c^{e(F)}; since c > 1 and c^{e(F)-e(H)} = (h^h(e(F)-e(H))+1)^{e(F)-e(H)} is vastly larger than h^{-h}, the bound is not derived. For the example H = K_3 and F = K_4, h=3, e(H)=3, e(F)=6, c=82, and the deletion term 82^6 n^{h - ...} dominates 82^3/27 n^{h - ...}. A correct argument would require, for instance, choosing c < h^{-h/(e(F)-e(H))}, e.g., c = 1/4 for this example. Since this proposition is presented as the representative application of the probabilistic method, this is a load-bearing defect in the survey's methodological content. The cited result [98] is likely correct, but the survey's derivation does not establish it.
  2. [Sections 2.1 and 4.2/4.5 (references [92] and [96])] The survey's central claim is that it provides a trustworthy and exhaustive inventory of the literature, but it incorporates results from two works marked 'in preparation' into the main text. In particular, the characterization of pairs (H,F) with ex(n,H,F) = O(1) and the linearity-versus-constancy dichotomy in Section 2.1 are attributed to [92], which is not publicly available. Similarly, several references to 'Part II' [96] describe results that do not yet exist. These items cannot be checked by the reader and should either be removed, clearly marked as private communications or announced results, or the survey should state explicitly that these claims are provisional.
minor comments (4)
  1. [Section 5.3] There is a typo in 'Removal :emma (Lemma 5.5)' — it should read 'Removal Lemma'.
  2. [Section 2.1] The sentence 'there are instances where there is non-vertex F-free graph' contains a typo; it should be 'there is no n-vertex F-free graph'.
  3. [Throughout] The authors claim an 'exhaustive approach to the collection literature' in the Introduction but also state that they 'will not always state all results completely or precisely'; the tension between these two statements should be resolved explicitly, for example by clarifying that the collection aims to be exhaustive while some statements are abbreviated.
  4. [Section 2, Theorem 2.8] The comment that the condition χ(H) < χ(F) in Halfpap and Palmer [123] 'can be safely dropped' is correct and follows from the Removal Lemma: if an n-vertex graph has few copies of F, deleting o(n^2) edges destroys all copies of F while removing only o(n^{|V(H)|}) copies of H, so the claimed lower bound on N(F,G) follows by contradiction. This is not a defect.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the survey is an inventory, and its self-citations are not load-bearing.

full rationale

This is a survey and literature inventory, not a derivation of a new extremal theorem from fitted inputs. Theorems and propositions are cited to independent published work; the many papers authored by the survey authors are presented as external results, not as premises that presuppose the survey's own conclusions. The organizing definitions (k-Turán-good, F-Turán-stable, t-Füredi-good) are terminology, not smuggled ansätze. The one genuinely unproved strengthening is the note after Theorem 2.8: 'the statement in [123] includes the assumption that χ(H)<χ(F), but this can be safely dropped without impacting the conclusion.' This is asserted without a proof, but it is not circular; it follows from the Removal Lemma, and it is not used to derive the survey's other claims. Separately, the printed proof of Proposition 5.7 is internally inconsistent: with c=h^h(e(F)-e(H))+1, the deletion term n^f p^{e(F)} n^{h-2} exceeds the lower bound (n/h)^h p^{e(H)} by a constant power of c, so the displayed difference is negative and the claimed lower bound is not derived as written. That is a correctness defect in the exposition of a cited result, not a circular reduction of the theorem to its own input. No fitted parameter is renamed a prediction, and no load-bearing uniqueness theorem is imported from the authors' prior work. Therefore the paper has no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 3 invented entities

The survey introduces no fitted numbers: there are no free parameters. The statements it catalogs depend on the standard machinery of extremal graph theory (regularity lemma, flag algebras, stability theory, spectral bounds, algebraic constructions) and on one conditional conjecture (Erdős-Sós). The only new 'entities' are definitional graph classes (Turán-good, Turán-stable, Füredi-good) that organize the literature; none carry an independent falsifiable handle.

assumptions (4)
  • standard math Szemerédi's Regularity Lemma (Theorem 5.2)
    Invoked in Section 5.1 for the proof sketch of Theorem 2.2 and underpins many surveyed results on dense graphs.
  • standard math Erdős-Stone-Simonovits theorem (Theorem 1.4)
    The asymptotic benchmark for the non-degenerate case; stated in Section 1.2 and used throughout Section 2.
  • standard math Validity of the flag algebra calculus of Razborov
    Section 5.4 presents flag algebras as an accepted tool; surveyed results such as ex(n,C_5,K_3) rest on flag algebra computations.
  • domain assumption Erdős-Sós conjecture, conditional
    The bound on ex(n,K_r,T) from Gerbner, Methuku and Palmer [93] in Section 4.2 holds only for trees whose subtrees satisfy the Erdős-Sós conjecture; the survey records the result as conditional.
invented entities (3)
  • k-Turán-good and F-Turán-good graph classes
    purpose: Organize surveyed exact results by whether the Turán graph T(n,k-1) maximizes the number of copies of H in K_k-free or F-free graphs.
    Definitional taxonomy defined in Sections 1.1 and 3; it structures the survey but makes no falsifiable predictions.
  • (weakly) F-Turán-stable graph classes
    purpose: Capture approximate stability conditions that, via the stability method, imply exact extremal results.
    Definitional taxonomy in Sections 3.1 and 5.3; used to organize stability-based proofs, with no empirical content of its own.
  • t-Füredi-good graphs
    purpose: Identify graphs H for which ex(n,H,K_{2,t}) is asymptotically realized by the Füredi graph F(n,t).
    Definitional notion from Section 5.7 (attributed to Gerbner [74]) that names a class of lower-bound results.

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Cite this review

Pith. "Pith review of Survey of generalized Tur\'an problems -- counting subgraphs." pith.science (2026). https://pith.science/paper/P75FB3WW

@misc{pith2026250603418,
  author       = {Pith},
  title        = {Pith review of: Survey of generalized Tur\'an problems -- counting subgraphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P75FB3WW}},
  note         = {Machine review of arXiv:2506.03418}
}
abstract

For fixed graphs $H$ and $F$, the \emph{generalized Tur\'an number} $\mathrm{ex}(n,H,F)$ is the maximum possible number of copies of a subgraph $H$ in an $n$-vertex $F$-free graph. This article is a survey of this extremal function whose study was initiated in an influential 2016 article by Alon and Shikhelman (\emph{J. Combin. Theory, B}, {\bf 121}, 2016).

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Forward citations

Cited by 3 Pith papers

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