---
title: Induced Turán Numbers in Extremal Graphs
url: https://www.emergentmind.com/topics/induced-turan-numbers
type: topic
---

# Induced Turán Numbers in Extremal Graphs

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:
\[
\operatorname{ex}(n,\{H,F\text{-ind}\}) = \max \left\{ |E(G)| : |V(G)|=n,\, H \not\subseteq G,\, F \not\hookrightarrow_\mathrm{ind} G \right\}
\]
Here $H\not\subseteq G$ means no (not necessarily induced) subgraph of $G$ is isomorphic to $H$, and $F \not\hookrightarrow_\mathrm{ind} G$ means $G$ has no induced subgraph isomorphic to $F$ [1610.06521, 2105.12503, 2409.12875].

The construction generalizes both classic Turán numbers (when $F$ is trivial) and induced-subgraph questions (when $H$ is omitted). For bipartite $F$, the function $\operatorname{ex}(n,\{F\text{-ind}\})$ is trivial except on highly restricted classes; interest thus centers on the mixed regime forbidding $H$ as a subgraph and $F$ as an induced subgraph.

## 2. Main Theorems and Regimes: Asymptotic Results

A central structural result establishes that for fixed non-bipartite $H$ and any $F$ not an independent set or complete bipartite, the induced Turán number is asymptotically controlled by the chromatic numbers $\chi(H)$ and $\chi(F)$:
\[
\operatorname{ex}(n,\{H,F\text{-ind}\})
=
\begin{cases}
\left(1-\frac{1}{r}+o(1)\right)\binom{n}{2} & s>r \text{ or $F$ not complete multipartite} \\
\left(1-\frac{1}{s}+o(1)\right)\binom{n}{2} & s<r,\, F \text{ complete $(s{+}1)$-partite}
\end{cases}
\]
where $r = \chi(H) - 1$, $s = \chi(F) - 1$. Thus, the extremal structure is governed by the denser of the Turán graphs $T_r(n), T_s(n)$, with a sharp switch depending on $F$'s multipartiteness [2105.12503].

For bipartite settings, particularly host graphs $K_{n,n}$, analogous sharp thresholds govern the function $\operatorname{ex}(K_{n,n},\{K_{t,t},H\text{-ind}\})$, now controlled by VC-dimension or maximal degree parameters of $H$:
\[
\operatorname{ex}(K_{n,n},\{K_{t,t},H\text{-ind}\}) = o(n^{2-1/d})
\]
where $d$ bounds the degree or VC-dimension in one bipartition class [2401.11296]. 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 [2601.10548].
- **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 [2512.16398, 2601.10548].
- **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 [1610.06521, 2401.11296, 2504.19094].

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

## 4. Key Examples and Special Cases

- **Induced Complete Bipartite Exclusion**: Forbidding $K_{s,t}$ as an induced subgraph with additional subgraph constraints yields the same exponent as the classical Kővári–Sós–Turán theorem, provided $H$ is non-bipartite. Complete $s$-partite graphs give constructions achieving the upper bound [1610.06521].
- **Odd Cycles and Induced Bicliques**: For $C_{2k+1}$-free graphs with no induced $K_{s,t}$, extremal numbers asymptotically match those for the non-induced case except in the exceptional case $(k,s,t)=(2,2,2)$ where an induced constraint strictly increases the extremal number [1707.06482].
- **Graphs with Constant Link**: Exact linear edge bounds are established for $\operatorname{ex}(n;C_k,K_{1,t}\text{-ind})$, connecting induced Turán problems with the structure of graphs whose neighborhoods induce fixed subgraphs [2409.12875].

### Table: Asymptotic Regimes for $\operatorname{ex}(n,\{H, F\text{-ind}\})$

| Regime                                                 | Asymptotic | Extremal construction                    |
|--------------------------------------------------------|------------|------------------------------------------|
| $H$ non-bipartite, $F$ not ind. set nor bipartite      | $(1-1/r)\binom{n}{2}$ or $(1-1/s)\binom{n}{2}$ | Turán graph $T_r(n)$ or $T_s(n)$         |
| Host $K_{n,n}$, $F=K_{t,t}$, bipartite $H$ of $\deg \le d$ | $o(n^{2-1/d})$ | Extreme bipartite, bounded-degree graphs |

## 5. Induced Turán Numbers in $K_{s,s}$-Free Settings and Rational Exponents

Recent developments have transferred classical bipartite extremal exponents to the induced setting via $K_{s,s}$-free constraints. For every rational $q\in(1,2)$, there exists a small forbidden induced family for which the induced Turán number grows as $\Theta(n^q)$. This is demonstrated by a supersaturation argument for induced trees and cycles in $K_{s,s}$-free graphs, and explicit constructions arising from appropriately lifted trees and cycles [2506.09020]. 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 $K_{s,s}$-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 [2506.09020].

## 6. Inducibility of Turán Graphs and Graphons

Inducibility questions ask for the maximum induced density of a fixed target graph $F$ in large $n$-vertex graphs, as $n\to\infty$. For Turán graphs $T(s,r)$ and more generally complete multipartite graphs, the extremal inducibility is realized by balanced multipartite graphs—and can be computed explicitly:
\[
i(T(s,r)) = g(\ell) = \frac{\ell!s!}{(\ell - r)!(r - q)!q!(p!)^r(p+1)^q\ell^s}
\]
where $s=pr+q$, $0\le q<r$, and $\ell$ is the maximizing part number solving $(\ell/(\ell - r))(1-1/\ell)^s > 1$ [2512.16398].

This confirms and generalizes previous conjectures of Bollobás–Egawa–Harris–Jin and establishes "perfect stability": for large $n$, the unique extremal configuration is the balanced $\ell$-partite Turán graph $T_\ell(n)$ [2601.10548]. The same machinery extends to forbidden $K_{k+1}$ 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 $H$ and $s\geq |V(H)|$, $\operatorname{ex}(n,\{K_{s,s},H\text{-ind}\}) = \Theta(\operatorname{ex}(n,H))$. The best known bounds match only in special cases and with suboptimal exponents; finding tight bounds or potential counterexamples is a leading direction [2504.19094].
- **Optimal Constants**: For classical forbidden pairs $(H,F)$ (e.g., $H$ non-bipartite, $F=K_{s,t}$), the optimal constants in asymptotic upper/lower bounds are not known even in the induced context [1610.06521].
- **Hypergraph Extensions**: The relationship between induced Turán numbers for $r$-uniform hypergraphs and classical clique Turán numbers via shadow and trace techniques remains incompletely understood, especially in the intermediate regime when $F$ is neither planar nor outerplanar [2002.07350].

Recent work employs dynamic programming and cotree decompositions to classify extremal cographs with forbidden bipartite patterns [2601.07406], 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.

Source: https://www.emergentmind.com/topics/induced-turan-numbers