Induced Turán Numbers in Extremal Graphs

• Induced Turán numbers are defined as the maximum edge count in an n-vertex graph that avoids a given subgraph H and an induced copy of F.
• They interpolate between classical Turán problems and induced subgraph constraints using techniques such as graph regularity, density arguments, and hypergraph methods.
• Recent results reveal asymptotic regimes driven by chromatic numbers and extend classical bounds to bipartite settings and graphs with rational exponents.

Induced Turán numbers generalize classical extremal problems by maximizing the number of edges in an $n$-vertex graph subject to simultaneously forbidding a given (not necessarily induced) subgraph $H$ and an induced copy of another subgraph $F$. This framework interpolates between classic Turán-type questions, induced subgraph constraints, and more subtle density phenomena in extremal combinatorics. Over the past decade, a systematic theory of induced Turán numbers has emerged, leveraging the interplay between forbidden substructures, graph regularity, and hypergraph machinery.

1. Formal Definition and Fundamental Examples

For fixed simple graphs $H$ and $F$, the induced Turán number $\operatorname{ex}(n,\{H,F\text{-ind}\})$ is the maximum number of edges in an $n$-vertex graph $G$ excluding $H$ as a subgraph and $F$ as an induced subgraph: $H$0 Here $H$1 means no (not necessarily induced) subgraph of $H$2 is isomorphic to $H$3, and $H$4 means $H$5 has no induced subgraph isomorphic to $H$6 (Loh et al., 2016, Illingworth, 2021, Caro et al., 2024).

The construction generalizes both classic Turán numbers (when $H$7 is trivial) and induced-subgraph questions (when $H$8 is omitted). For bipartite $H$9, the function $F$0 is trivial except on highly restricted classes; interest thus centers on the mixed regime forbidding $F$1 as a subgraph and $F$2 as an induced subgraph.

2. Main Theorems and Regimes: Asymptotic Results

A central structural result establishes that for fixed non-bipartite $F$3 and any $F$4 not an independent set or complete bipartite, the induced Turán number is asymptotically controlled by the chromatic numbers $F$5 and $F$6: $F$7 where $F$8, $F$9. Thus, the extremal structure is governed by the denser of the Turán graphs $H$0, with a sharp switch depending on $H$1's multipartiteness (Illingworth, 2021).

For bipartite settings, particularly host graphs $H$2, analogous sharp thresholds govern the function $H$3, now controlled by VC-dimension or maximal degree parameters of $H$4: $H$5 where $H$6 bounds the degree or VC-dimension in one bipartition class (Axenovich et al., 2024). This result strengthens previous bipartite bounds and elucidates the tight connection with classic Zarankiewicz-type theorems.

3. General Structural and Proof Techniques

The theory combines symmetrization, density arguments, and combinatorial optimization:

• Zykov Symmetrization ensures that for complete multipartite forbidden patterns, extremal graphs achieving the maximum are themselves multipartite (Liu et al., 15 Jan 2026).
• Density Increment and Regularity: For many parameter ranges, max-density is uniquely achieved by Turán-type graphs, and the extremal problem reduces to a density optimization over possible part sizes in equipartite graphons (Yuster, 18 Dec 2025, Liu et al., 15 Jan 2026).
• Dependent Random Choice and Ramsey Theory: In bipartite or more intricate regimes, DRC and Ramsey-theoretic arguments are used to force either the forbidden induced structure or large homogeneous sets (Loh et al., 2016, Axenovich et al., 2024, Sheffield, 27 Apr 2025).

Further, explicit combinatorial constructions and stability results yield uniqueness of extremal graphs and fine asymptotic expansions for large $H$7.

4. Key Examples and Special Cases

• Induced Complete Bipartite Exclusion: Forbidding $H$8 as an induced subgraph with additional subgraph constraints yields the same exponent as the classical Kővári–Sós–Turán theorem, provided $H$9 is non-bipartite. Complete $F$0-partite graphs give constructions achieving the upper bound (Loh et al., 2016).
• Odd Cycles and Induced Bicliques: For $F$1-free graphs with no induced $F$2, extremal numbers asymptotically match those for the non-induced case except in the exceptional case $F$3 where an induced constraint strictly increases the extremal number (Ergemlidze et al., 2017).
• Graphs with Constant Link: Exact linear edge bounds are established for $F$4, connecting induced Turán problems with the structure of graphs whose neighborhoods induce fixed subgraphs (Caro et al., 2024).

Table: Asymptotic Regimes for $F$5

Regime	Asymptotic	Extremal construction
$F$6 non-bipartite, $F$7 not ind. set nor bipartite	$F$8 or $F$9	Turán graph $\operatorname{ex}(n,\{H,F\text{-ind}\})$0 or $\operatorname{ex}(n,\{H,F\text{-ind}\})$1
Host $\operatorname{ex}(n,\{H,F\text{-ind}\})$2, $\operatorname{ex}(n,\{H,F\text{-ind}\})$3, bipartite $\operatorname{ex}(n,\{H,F\text{-ind}\})$4 of $\operatorname{ex}(n,\{H,F\text{-ind}\})$5	$\operatorname{ex}(n,\{H,F\text{-ind}\})$6	Extreme bipartite, bounded-degree graphs

5. Induced Turán Numbers in $\operatorname{ex}(n,\{H,F\text{-ind}\})$7-Free Settings and Rational Exponents

Recent developments have transferred classical bipartite extremal exponents to the induced setting via $\operatorname{ex}(n,\{H,F\text{-ind}\})$8-free constraints. For every rational $\operatorname{ex}(n,\{H,F\text{-ind}\})$9, there exists a small forbidden induced family for which the induced Turán number grows as $n$0. This is demonstrated by a supersaturation argument for induced trees and cycles in $n$1-free graphs, and explicit constructions arising from appropriately lifted trees and cycles (Dong et al., 10 Jun 2025). This universality in realizing rational exponents substantially extends the paradigm launched by the Bukh–Conlon random algebraic construction.

In addition, induced extremal numbers for thetas and prism graphs in $n$2-free hosts sharply match classical lower bounds up to poly-log factors, further supporting the conjecture that induced and classical bipartite exponents match up to constants (Dong et al., 10 Jun 2025).

6. Inducibility of Turán Graphs and Graphons

Inducibility questions ask for the maximum induced density of a fixed target graph $n$3 in large $n$4-vertex graphs, as $n$5. For Turán graphs $n$6 and more generally complete multipartite graphs, the extremal inducibility is realized by balanced multipartite graphs—and can be computed explicitly: $n$7 where $n$8, $n$9, and $G$0 is the maximizing part number solving $G$1 (Yuster, 18 Dec 2025).

This confirms and generalizes previous conjectures of Bollobás–Egawa–Harris–Jin and establishes "perfect stability": for large $G$2, the unique extremal configuration is the balanced $G$3-partite Turán graph $G$4 (Liu et al., 15 Jan 2026). The same machinery extends to forbidden $G$5 situations, and to bipartite settings and induced matching/triangle-factor densities.

7. Open Problems and Future Directions

Several conjectures remain central to the field:

• Hunter–Milojević–Sudakov–Tomon Conjecture: For all bipartite $G$6 and $G$7, $G$8. The best known bounds match only in special cases and with suboptimal exponents; finding tight bounds or potential counterexamples is a leading direction (Sheffield, 27 Apr 2025).
• Optimal Constants: For classical forbidden pairs $G$9 (e.g., $H$0 non-bipartite, $H$1), the optimal constants in asymptotic upper/lower bounds are not known even in the induced context (Loh et al., 2016).
• Hypergraph Extensions: The relationship between induced Turán numbers for $H$2-uniform hypergraphs and classical clique Turán numbers via shadow and trace techniques remains incompletely understood, especially in the intermediate regime when $H$3 is neither planar nor outerplanar (Furedi et al., 2020).

Recent work employs dynamic programming and cotree decompositions to classify extremal cographs with forbidden bipartite patterns (Zimmermann, 12 Jan 2026), and further methodological developments in regularization, dependent random choice, and spectral extremality are expected to deepen the theory.

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The induced Turán function unifies diverse phenomena in extremal combinatorics, revealing a nuanced interplay between forbidden induced substructure, global density, and multipartite extremal configurations. The field continues to connect combinatorial optimization, Ramsey theory, and hypergraph methods, with many central questions open.