Linear Turán Numbers

• Linear Turán numbers are extremal functions for linear r-uniform hypergraphs, measuring maximum edge density under forbidden configuration constraints.
• They extend classical Turán problems from graphs to hypergraphs, combining combinatorial design, stability analysis, and explicit construction techniques.
• Key examples include linear paths, crowns, and fans, with results obtained via vertex weighting, closure operations, and decomposition strategies.

A linear Turán number is an extremal function for linear uniform hypergraphs and their forbidden substructures, quantifying the maximal edge density under linearity and exclusion constraints. The study of linear Turán numbers extends classical Turán-type extremal problems from graphs to hypergraphs, with a strong emphasis on combinatorial design, stability, and exact constructions for small forbidden configurations.

1. Definitions and General Framework

For fixed integers $r\ge2$ and $n\ge1$, a hypergraph $H=(V,E)$ is $r$-uniform if every edge $e\in E$ has cardinality $r$. $H$ is linear if every pair of distinct edges intersects in at most one vertex. Given a family $\mathcal F$ of linear $r$-graphs, the linear Turán number is defined as

$\ex_r^{\lin}(n, \mathcal F) = \max\left\{\,|E(H)| : |V(H)|=n,\ H\text{ linear, } H\text{ is } \mathcal F\text{-free}\,\right\}$

where $H$ is $\mathcal F$-free if it contains no member of $\mathcal F$ as a (not necessarily induced) subhypergraph. When $\mathcal F = \{F\}$ is a singleton, write $\ex_r^{\lin}(n,F)$. This quantity generalizes the classical Turán number by incorporating the linearity restriction and often leads to different extremal behavior.

2. Linear Turán Numbers in Graphs and Hypergraphs

For $r=2$ (i.e., graphs), linearity is trivial, so the linear Turán number coincides with the traditional version. In this case, for a family $\mathcal F$ of graphs,

$\ex^{\lin}_2(n,\mathcal F) = \ex(n,\mathcal F)$

For $r\ge3$, new combinatorial phenomena arise due to the interplay of uniformity and linearity. Prototypical forbidden configurations include linear paths, cycles, matchings, fans, crowns, and more complex tree-like or acyclic substructures.

3. Prototypical Results and Extremal Constructions

Results for linear Turán numbers often combine explicit extremal constructions with sharp upper bounds via structural reducibility, neighborhood/degree methods, and sometimes averaging or "weight-transfer" arguments.

3.1. Linear Paths and Trees

For a linear $r$-uniform path of length $\ell$, $L_{\ell}^{(r)}$, the Turán number for $k$ vertex-disjoint copies is (Bushaw et al., 2013):

$$ex_r\left(n; k \cdot L^{(r)}_{\ell}\right) = \sum_{i=1}^{t_k} \binom{n-i}{r-1} + d_\ell$$

with $t_k = k\lfloor(\ell+1)/2\rfloor - 1$ and an explicit $d_\ell$ depending on parity.

For $r=3$, the exact extremal constructions are often based on blow-ups, steered by blocking sets (fixed vertex sets $S$) such that all edges intersect $S$. For small tree-like configurations, exact values or tight asymptotic bounds are known, often exploiting the structure of relevant Steiner systems (Adak et al., 24 Jan 2026).

3.2. Crowns, Fans, and Small Configurations

The crown $C_{13}$ is the 3-graph on 9 vertices with four edges in a cyclic, crown-like arrangement. The sharp bound is (Tang et al., 2021):

$|E(G)| \leq \frac{3(n-s)}{2}$

where $s$ is the number of vertices of degree at least 6. This result is tight and completes the classification of linear Turán numbers for all 3-graphs with at most 4 edges.

The $k$-fan $F^k$ in $k$-uniform linear hypergraphs satisfies

$ex_{\rm lin}(n, F^k) \leq \frac{n^2}{k^2}$

and equality holds if and only if $n$ is divisible by $k$ and the extremal example is a transversal design (Füredi et al., 2017).

3.3. Linear Cycles

For linear 3-uniform cycles:

• The 5-cycle: $\ex^{\lin}(n, C_5) \sim \frac{1}{3\sqrt{3}} n^{3/2}$ (Ergemlidze et al., 2017).
• For general odd length $2k+1$: $\ex^{\lin}(n, C_{2k+1}) = \Theta(n^{1+1/k})$ for $k=2,3,4,6$.

Upper bounds typically leverage reductions to extremal problems for high-girth bipartite graphs.

4. Structural and Proof Techniques

Techniques to determine or bound $\ex_r^{\lin}(n,F)$ include:

• Vertex weighting and averaging: Assigning weights selectively to vertices of bounded degree and averaging over edges to obtain contradictions if the bound is exceeded (Tang et al., 2021).
• Closure operations: Applying stable closure operations (e.g., $k$-closure in graphs) to guarantee that certain structural invariants persist, enabling induction or reduction (Ning et al., 2018).
• Decomposition into small components: Deleting subgraphs that exhibit rigid neighborhood structure, especially in proofs for small forbidden configurations.
• Degree and link-graph analysis: Using maximal degree constraints and properties of associated link graphs (e.g., requiring matching or cycle structures) to forbid forbidden subgraphs (Fletcher, 2021).
• Explicit designs: Constructing extremal examples via transversal designs, Steiner systems, or combinatorial block systems for both sharp lower bounds and characterizations of equality (Füredi et al., 2017, Adak et al., 24 Jan 2026).

5. Exact Formulas and Classification for Special Cases

For linear forests in graphs (vertex-disjoint union of paths), the exact Turán number is (Ning et al., 2018):

$$ex(n; \mathcal{L}_{n,k}) = \max\left\{\binom{k}{2},\; \binom{n}{2} - \binom{n-\lfloor(k-1)/2\rfloor}{2} + c\right\}, \qquad c = \begin{cases} 0 & \text{if } k \text{ odd} \ 1 & \text{if } k \text{ even} \end{cases}$$

with explicit constructions providing sharpness: a clique on $k$ vertices plus isolated vertices, or a join of a small clique with an independent set (Ning et al., 2018).

In 3-uniform linear hypergraphs, the Turán numbers for the 3-fan and the crown have been determined up to exact constants and structural characterization of extremal systems (Ergemlidze et al., 2020, Fletcher, 2021, Tang et al., 2021).

6. Broader Context and Open Problems

The linear Turán number paradigm helps bridge extremal graph theory, design theory, and probabilistic combinatorics. Open directions include:

• Sharp asymptotics or exact values for longer linear paths and trees in $r$-uniform systems for $r>3$ (Adak et al., 24 Jan 2026).
• Classification of extremal constructions beyond small parameters, particularly for configurations where the block structure is not governed by classical transversal designs or Steiner systems.
• Resolution of the precise order of magnitude and leading constants for linear Turán numbers of specific acyclic or cyclic configurations (e.g., higher arity crowns, cycles, and paths).

The methodology continues to yield robust stability results, transfer theorems, and rich structural descriptions for extremal hypergraph families. This underscores the centrality of linear Turán numbers in contemporary extremal combinatorics and their intersection with finite geometry and design theory (Ning et al., 2018, Tang et al., 2021, Füredi et al., 2017, Ergemlidze et al., 2017, Adak et al., 24 Jan 2026).