Generalized Turán Numbers for Expansions

• The paper establishes exact non-degenerate and asymptotic results for ex_r(n, K_s^(r), F^(r)) using balanced multipartite constructions.
• It employs density dichotomy, Lagrangian methods, and stability arguments to derive sharp bounds for r-expansions based on the chromatic properties of the base graph.
• The research bridges classical Turán theory and modern hypergraph design, offering new insights into asymptotic, degenerate, and probabilistic regimes.

A generalized Turán number for expansions concerns the extremal function ex_r(n, 𝓗, 𝓕), defined as the maximum number of copies of an r-uniform hypergraph 𝓗 in an n-vertex r-uniform hypergraph G that contains no copy of a fixed r-uniform hypergraph 𝓕. In the context of expansions, the focus is on the case where 𝓕 is the r-expansion of a graph F, constructed by replacing each edge of F by a unique r-element set containing that edge and r − 2 new, pairwise disjoint vertices. This area synthesizes Turán-type extremal combinatorics for graphs and hypergraphs, with particular emphasis on how the chromatic and structural properties of F govern the order of magnitude and critical exponents in the extremal function.

1. Foundational Definitions and Problem Setting

Given a graph F and an integer r ≥ 2, the r-expansion F^{(r)} is realized by mapping each edge {u, v} of F to the r-set {u, v} ∪ Y_e, where Y_e consists of r − 2 distinct vertices disjoint from V(F) and from all other Y_e′. More generally, for r-uniform hypergraphs 𝓗 and 𝓕, the generalized Turán number is

$\ex_r(n, 𝓗, 𝓕)=\max\{|E_{𝓗}(G)|: G\subseteq K^{(r)}_n,\ G \text{ does not contain } 𝓕\}$

where |E_{𝓗}(G)| denotes the number of (labeled) copies of 𝓗 in G. The classical Turán number is recovered as ex_r(n, K_2^{(r)}, 𝓕)=ex_r(n, 𝓕) (the maximum number of edges).

The study of ex_r(n, 𝓗, 𝓕) for expansion families 𝓕=F^{(r)} generalizes multiple lines of classical extremal graph and hypergraph theory, including Turán-type theorems for complete and bipartite graphs, paths, cycles, and more structurally complex subgraphs. Notably, the case r=2 is the setting of the generalized Turán problem of Alon–Shikhelman.

2. Exact Results for Expansions of Complete Graphs and Non-degenerate Cases

For ex_r(n, K_s^{(r)}, F^{(r)}), the key dichotomy is controlled by the chromatic number χ(F). The non-degenerate regime (Θ(n^s)) arises precisely when χ(F)>s. In this case, every F^{(r)}-free configuration is necessarily contained in an ℓ-partite r-graph (ℓ=χ(F)−1); the extremal construction is a balanced ℓ-partite, complete r-graph _r(n,ℓ).

The main results for this regime, as established in "On generalized Turán problems for expansions" (Zhou et al., 14 Jan 2026), are:

• Theorem A (Exact non-degenerate): For r≥3, s≤ℓ, all large n,

$\ex_r(n, K_s^{(r)}, K_{\ell+1}^{(r)}) = \#(s\text{-cliques in }_r(n, ℓ))$

and this is attained uniquely by the balanced ℓ-partite r-graph.

• Theorem B (Union of cliques, asymptotic): For vertex-disjoint unions of complete graphs (kK_{ℓ+1}^{(r)}), the extremal structure and enumeration are determined by a “Turán-plus-exceptional-set” construction, generalizing both the Zykov–Erdős–Turán theory and union-of-cliques for r=2.

This dichotomy matches — and, for r≥3, extends — the classical threshold for ex(n,K_s,F) in graphs (Zhou et al., 14 Jan 2026).

3. Degenerate Cases: Forests, Paths, Stars, and Structural Thresholds

When χ(F)≤s, the problem is degenerate: ex_r(n, K_s^{(r)}, F^{(r)})=o(n^s). The order of magnitude and leading exponents depend delicately on the forbidden expansion family F^{(r)} and the host substructure.

For forests and related classes:

• For a single star S_ℓ^{(r)} (the r-expansion of an ℓ-edge star):
  • If s≤ℓ+r−2,

  $ex_r(n, K_s^{(r)}, S_ℓ^{(r)}) = Θ(n^{r−2})$

  with the upper bound matched by a construction based on the “shadow” hypergraph formalism (Zhou et al., 14 Jan 2026).

• For star-forests, linear forests (unions of paths P_{ℓ_i}^{(r)}) and star-path forests, precise asymptotics or at least exponents are given (see Theorems 3.2–3.4 in (Zhou et al., 14 Jan 2026)). The transition between Θ(n^{r−2}) and Θ(n^{r−1}) reflects the combinatorial sparsity (star-type) or “pathlike” structure.
• For expansions of small graphs with higher connectivity, such as triangles or cycles, extremal configurations and densities draw on design theory and stability analysis. For instance, the extremal function ex(n,T_r) for generalized triangles is exactly attained by balanced blow-ups of the unique Steiner system S_r, at least for r∈{5,6} (Norin et al., 2015).

4. Stability, Extremal Constructions, and the Lagrangian Perspective

The stability method is central in elevating density/averaging theorems to exact Turán numbers for expansion families. The proof machinery in (Norin et al., 2015)—which is archetypal—proceeds via:

1. Density Dichotomy and Lagrangian Methods: For a given expansion or extension hypergraph, the Turán density is given by the Lagrangian of a certain partite r-graph (often a balanced blow-up).
2. Vertex-local and Global Stability: Adaptations of classical Erdős–Simonovits stability, via local link graphs, removal lemmas, and symmetrization, establish that extremal configurations must be close (in edit-distance) to a unique partite blow-up.
3. Exactness via Bootstrap: Once local and weighted stability are established, a general compactness/bootstrapping theorem yields uniqueness and exactness for all sufficiently large n.

When the forbidden core configuration is edge-critical, the error terms vanish and extremal numbers coincide with the balanced Turán number (see (Tang et al., 2019)): for such F, ex(n,H_F^{(r)}) = t_r(n,ℓ−1), with t_r(n,ℓ−1) the number of edges in a balanced (ℓ−1)-partite r-graph.

5. Expansions in Sparse Random Hypergraphs and Supersaturation

Random Turán-type theorems for expansions are developed in (Nie, 2023). For the random r-uniform hypergraph G^r_{n,p}, the random Turán number ex(G^r_{n,p}, H^{(+r)}) exhibits phase transitions controlled by spreadness parameters s(F) linked to the structure of the expanded template. The techniques rely on balanced supersaturation via codegree dichotomy and the hypergraph container method.

• For expansions of K_{k}^{k−1} (the complete (k−1)-graph on k vertices), the bound is Θ(p n^{r−1}) in high density, and sharp exponents in “middle” regimes, with matching lower bounds from blow-ups of Gowers–Janzer constructions.
• For expansions of tight trees with empty total intersection, Θ(p n^{r−1}) persists in the dense regime, with transitions to lower exponents as p decreases, captured by star constructions and spreadness bounds (Propositions 3.1–3.2 in (Nie, 2023)).
• The balanced supersaturation argument employs recursive partitioning by codegree, greedy expansion, and an application of hypergraph containers to extract tight upper bounds even in the sparse regime.

6. Core Open Problems, Conjectures, and Structural Transitions

The expansion problem is not fully classified for general base graphs. For instance, the precise conditions for quadratic versus superquadratic growth in ex_3(n,G^+) remain open outside the known golden-ratio and tree-width-2 criteria (Kostochka et al., 2014). Moreover, the search for uniqueness of extremal constructions, especially when Steiner systems or covering designs are not unique or do not exist, is ongoing (Norin et al., 2015).

Major open problems include:

• Determination of exact constants and extremal structures for expansions of various graphs beyond the complete or bipartite case.
• Classification of which expanded graphs yield quadratic, subquadratic, or polynomial but superquadratic extremal functions.
• Understanding structural thresholds in chromatic number or tree-width that lead to distinct asymptotic regimes.

Several conjectures, such as quadratic growth for expansions of all graphs of tree-width ≤2, represent natural frontiers (Kostochka et al., 2014).

7. Connections to Classical and Recent Advances

The generalized Turán numbers for expansions unify and generalize a range of classical combinatorial results:

• For r=2, the problem reduces to the classical graph theory of Alon–Shikhelman (2016), Zykov (1949), Erdős (1962), and their modern extensions (Zhou et al., 14 Jan 2026).
• The extension approach and Lagrangian methods highlight deep connections between extremal hypergraph theory, stability phenomena, and design theory (notably through unique dense Steiner systems (Norin et al., 2015)).
• The error term and decomposition family approach in expansion theorems (biex(n,F)) builds on and sharpens constructions going back to Pikhurko, Mubayi, and others (Tang et al., 2019).

In summary, the study of generalized Turán numbers for expansions provides a robust framework for extremal problems involving forbidden expanded substructures, revealing a compelling interplay between chromatic thresholds, expansion parameters, stability, and probabilistic techniques. The landscape is both rich and evolving, combining foundational combinatorics, probabilistic arguments, and algebraic constructions to resolve old questions and pose new directions.