Generalized Turán Number

• Generalized Turán numbers are functions that count the maximum labeled copies of a subgraph H in an n-vertex graph that avoids a forbidden graph F.
• They extend classical Turán theory by considering arbitrary subgraphs and forbidden configurations, using methods such as regularity, shifting, and blow-up constructions.
• These numbers have broad implications in extremal combinatorics, offering insights into graph structure, coloring, and stability in extremal graphs.

A generalized Turán number is a graph extremal function that counts, for fixed graphs $H$ and $F$, the maximum number of (labeled) copies of $H$ in an $n$-vertex graph that does not contain a copy of $F$. This central concept significantly extends the classical Turán theory (which considers the maximum number of edges in an $F$-free graph) by allowing arbitrary subgraphs $H$ to be counted and arbitrary $F$ to be forbidden, and thus probes deeper into extremal combinatorics and its relation to graph structure, coloring, and forbidden configurations.

1. Formal Definition and Historical Context

Let $H$ and $F$ be simple graphs. The generalized Turán number is defined as

$$\mathrm{ex}(n,H,F) = \max\{ N(H, G) : |V(G)|=n,\, F\not\subseteq G \}$$

where $N(H, G)$ denotes the number of (labeled) copies of $H$ in $G$. When $H=K_2$, this specializes to the classical Turán number $\mathrm{ex}(n,F)$, the maximum number of edges in an $n$-vertex $F$-free graph.

The foundational work by Erdős and Turán established extremal numbers for complete graphs, with the Turán graph being extremal for $\mathrm{ex}(n, K_r)$ when forbidding $K_{r+1}$. Alon and Shikhelman (Ma et al., 2018) introduced the broader family $\mathrm{ex}(n, T, H)$ for arbitrary $(T, H)$, extending the classical theorems and enriching the range of extremal questions in graph theory.

2. Structural Principles and Main Theorems

Chromatic Threshold and Turán-Type Stability

A critical regime is when the chromatic number of $T$ is less than that of $H$, $\chi(T) < \chi(H)$, the "non-degenerate" regime. Here, the density of the extremal host is high—skirting the threshold for forbidden substructures controlled by the coloring constraints. Classical results, such as Turán’s theorem and Erdős’s generalization, assert that the balanced complete $r$-partite graph $T_r(n)$ maximizes the number of $m$-cliques in any $K_{r+1}$-free $n$-vertex graph (Ma et al., 2018).

For arbitrary forbidden graphs $H$ with $\chi(H)=r+1$, asymptotically

$$\mathrm{ex}(n, K_m, H) = \binom{r}{m} \left( \frac{n}{r} \right)^m + o(n^m)$$

and finer results establish the error term $o(n^m)$ up to a constant factor, depending on the decomposition family $\mathcal{F}_H$ related to $H$: $$\mathrm{ex}(n, K_m, H) = \binom{r}{m} \left( \frac{n}{r} \right)^m + \Theta(\mathrm{biex}(n, H) n^{m-2})$$ where $\mathrm{biex}(n, H)$ is the Turán number for the family of bipartite subgraphs derived from color class deletions in a proper coloring of $H$ (Ma et al., 2018).

Stability theorems ensure that almost extremal graphs (those close to maximizing $N(H, G)$) are structurally close to the Turán construction unless special algebraic or local augmentations intervene (Gerbner, 2022).

3. Methodological Highlights and Proof Techniques

(i) Reduction to Controlled Subgraphs:

Key reductions employ degree control, extracting large minimum-degree subgraphs—often bipartite—whose extremal structure can be analyzed or regularized (Liu et al., 2021).

(ii) Regularity and Tree-embedding Techniques:

For bipartite forbidden structures (e.g., cycles, theta-graphs), arguments employ BFS layering, extraction of almost-tree subgraphs, and custom supersaturation bounds to show that excessive local density would force an excluded configuration (Liu et al., 2021).

(iii) Shifting Method:

For matchings or linear forests as forbidden graphs, the shifting method is used to optimize the structure of the extremal graph, leading to template graphs. This exploits the monotonicity of the shifting operation to reduce to a finite family of candidates for extremality (Wang, 2018).

(iv) Blow-up and Join Constructions:

For multiple forbidden subgraphs or expansions in the hypergraph setting, extremal graphs are often constructed by taking joins of carefully chosen subgraphs (e.g., $K_t \vee H$ for forbidding multiple copies or products) or by blow-ups of small dense blocks (Yang et al., 8 Aug 2025, Gerbner, 2023).

(v) Partial Blow-up and Core Lemmas:

Counting arbitrary subgraphs with forbidden matching or other degenerate structures is often resolved by identifying subsets of vertices ("blow-up cores") whose duplication does not create forbidden subgraphs but increases the number of countable subgraphs (Gerbner, 2023, Gerbner, 2023).

4. Key Results for Special Graph Classes

Forbidding	Counting	Leading Order/Formula	Extremal Structure
$K_{r+1}$	$K_m$	$\binom{r}{m}(n/r)^m + o(n^m)$	Balanced $r$-partite Turán graph $T_r(n)$
$M_{k+1}$	$K_s$, $K^*_{s,t}$	Maximum of clique/join counts [see explicit formulas]	Clique, or join $K_k \vee E_{n-k}$ (Wang, 2018)
$tK_r$	$K_s$	Explicit join construction; order-of-magnitude $n^{b(H)}$	Clique joined to Turán partite, as in (Gerbner, 2023)
linear forest $F$	$K_s$	Explicit (see (Zhu et al., 2021, Zhang et al., 2020)); two-term maximum	Clique or clique joined to independent set
planar graphs	trees, cycles, subgraphs	Exponential in blow-up index, often $\Theta(n^\alpha)$	Blow-ups in bounded degree/minor-closed substructures
matching number	$H$	$\Theta(n^{b(H,s)})$, explicit template constructions	Core plus independent set (Gerbner, 2023, Gerbner, 2024)

5. Special Topics: Generalized Turán Numbers for Specific Graphs and Regimes

Generalized Theta Graphs

For the generalized theta $\Theta_{k_1,\ldots,k_\ell}$, defined as the graph formed by joining two vertices with $\ell$ pairwise internally disjoint paths of lengths $k_i$, the best-known bound for the extremal number is

$$\mathrm{ex}(n, \Theta_{k_1, \ldots, k_\ell}) = O(n^{1+1/k^*})$$

where $k^* = \frac12 \min_{i<j}(k_i + k_j)$, with tight bounds in certain cases (Liu et al., 2021).

Disjoint Forbidden Subgraphs

For forbidding $t+1$ disjoint copies of a degenerate graph (e.g., $(t+1)K_{a,b}$), the extremal count for $K_r$ is

$$\sum_{s=0}^r \binom{t}{r-s} \frac{1}{s!} (n-t)^{s - \frac{s(s-1)}{2a}} + o(n^{\text{exp}})$$

and similar forms for disjoint cycles (Yang et al., 8 Aug 2025).

Regular Host Graphs

A further generalization considers the number of $H$ in $F$-free regular graphs. For many parameters, the extremal count matches the optimal value in the unrestricted case, but in others regularity imposes different asymptotics or sharp thresholds (Gerbner et al., 2023).

Planar, Minor-closed, and Outerplanar Cases

In minor-closed host graph families, the order of growth is often integer-powered in $n$ and governed by local blow-up structures (the "blow-up index") which measure the extent to which components can be independently enlarged without destroying planarity or another minor-closed property (Győri et al., 2020, Győri et al., 2021).

6. Open Problems and Conjectures

Several central questions remain in generalized Turán theory:

• Determining exact (not just asymptotic) extremal numbers and structures for arbitrary $(H,F)$ pairs and for host graphs with additional constraints (planarity, regularity, minor-exclusion).
• Understanding for which rational $\alpha \in (1,2)$ there exists a bipartite $F$ with $\mathrm{ex}(n,F) = \Theta(n^\alpha)$ (the "rational exponent" conjecture) (Liu et al., 2021).
• Extending classification to forbidden hypergraph substructures and their expansions, including the chromatic threshold phenomena in those contexts (Zhou et al., 14 Jan 2026, Gerbner, 2023).
• Developing stability results that quantify proximity of near-extremal graphs to the canonical extremal constructions, especially beyond the classically colored or partite settings (Gerbner, 2022).

The field remains rich with direction for further exploration, blending structural, probabilistic, and algebraic methods.

7. Connections and Broader Implications

Generalized Turán numbers unify extremal problems across graph theory, including matching theory, forbidden subgraph/induced subgraph enumeration, hypergraph extremal functions, and planar/minor-closed structure. Their study has yielded structural decomposition lemmas, refined stability theorems, connections to coloring and spectrum theory, and nontrivial algebraic and geometric extremal constructions. The combinatorial techniques developed for generalized Turán theory have broad applicability throughout combinatorics, discrete geometry, and graph algorithms.

References:

For main expository results, see (Ma et al., 2018, Gerbner, 2022, Zhang et al., 2020, Wang, 2018, Liu et al., 2021, Yang et al., 8 Aug 2025, Gerbner, 2023, Zhou et al., 14 Jan 2026, Gerbner et al., 2021, Gerbner et al., 2023, Xue et al., 2022, Gerbner, 2024, Wang et al., 14 Jan 2026, Zhu et al., 2021).