Turán-Type Extremal Problems in Simplicial Complexes

• Turán-type extremal problems for simplicial complexes study the maximum number of facets an n-vertex complex can have without containing a forbidden homeomorphic subcomplex.
• Recent advances apply probabilistic techniques, balanced multipartite reductions, and spectral methods to tighten bounds, such as improving the exponent from 3-1/5 to 8/3 for 2-complexes.
• Significant findings include tight Theta(n^(5/2)) bounds for surfaces like the torus and projective plane, alongside open questions on extending these techniques to higher dimensions and topological invariants.

A Turán-type extremal problem for simplicial complexes seeks the maximum size of a complex on $n$ vertices—the extremal number—avoiding a fixed forbidden subcomplex, often up to homeomorphism. In the two-dimensional case, this is closely tied to topological, combinatorial, and spectral analogues of classical Turán-type extremal results from graph and hypergraph theory, and probes deep connections between topology, probabilistic combinatorics, and extremal structural theory. Recent advances have improved universal exponent bounds, clarified structure for special classes, and illuminated longstanding open questions regarding surfaces and spheres.

1. Foundational Definitions and Framework

A two-dimensional simplicial complex $\mathcal{K}$ is specified by a vertex set $V$, with edges $E \subseteq \binom{V}{2}$ and triangular faces (facets) $F \subseteq \binom{V}{3}$, closed under inclusion: every triangle must have all edges present. The number of facets is denoted $f_2(\mathcal{K}) = |F|$. Given a fixed 2-dimensional simplicial complex $S$, a homeomorphic copy ("homeomorph") in a complex $G$ is a subcomplex created by subdividing some edges and triangles so that the underlying topological spaces are homeomorphic; equivalently, there exists a continuous bijection between their geometric realizations with a continuous inverse.

The topological Turán number $\operatorname{ex}_{\mathrm{top}}(n,S)$ is the largest number of facets that an $n$-vertex 2-complex may have without containing a homeomorph of $S$ (Keevash et al., 2020). More generally, the extremal number $\operatorname{ex}(n,F)$ for a (possibly non-uniform) simplicial complex $F$ is the maximal number of faces in a labeled $n$-vertex complex without an isomorphic subcomplex to $F$ (strong embedding) (Conlon et al., 2023, Axenovich et al., 18 Aug 2025).

2. Universal Exponents and Main Theorems

Recent progress centers on determining sharp exponents for topological Turán numbers of 2-complexes. Keevash–Long–Narayanan–Scott established that for any fixed $S$ there exists $C_S > 0$ such that (Keevash et al., 2020)

$$\operatorname{ex}_{\mathrm{top}}(n,S) \leq C_S n^{3-1/5}$$

i.e., any $n$-vertex 2-complex with $\geq C_S n^{3-1/5}$ facets must contain a homeomorph of $S$. This marked the first general bound in response to a question of Linial (2006).

Later, a further improvement was achieved: for any 2-dimensional $X$, $$\operatorname{ex}_{\mathrm{hom}}(n,X) \leq C n^{8/3}$$, strictly improving the exponent $3-1/5$ (i.e., $2.8$) to $8/3 \approx 2.667$ (Sankar, 16 Aug 2024). For central surfaces such as the torus $\mathbb{T}^2$ and projective plane $\mathbb{RP}^2$, the true exponent is $\Theta(n^{5/2})$, and this tight bound extends to all closed orientable and non-orientable surfaces via connected-sum constructions.

The exponents can be summarized as follows:

Object / Class	Upper Bound	Lower Bound
Arbitrary 2-complex	$O(n^{8/3})$	Universal, tight for some $X$
Torus / RP$^2$	$O(n^{5/2})$	$\Theta(n^{5/2})$

For higher dimensions $d$ (e.g., spheres $S^d$), the current best bounds are $O(n^{d+1-1/2^{d-1}})$ and conditional lower bounds $\Omega(n^{d+1-(d+1)/(2^{d+1}-2)})$; the conjectured true exponent matches the conditional lower (Newman et al., 8 Mar 2024). For many concrete examples—e.g., downward-closed complexes or cycle/matching "sandwiches"—precise extremal numbers are determined via generating sets and symmetry (Axenovich et al., 18 Aug 2025, Conlon et al., 2023).

3. Combinatorial and Probabilistic Methodologies

Key proof techniques include reductions to balanced multipartite 3-uniform hypergraphs and application of dependent random choice for local density increments in link graphs. For instance, to embed homeomorphs, one identifies a vertex $z$ with dense link graph $L_z$, cleans bad pairs/triples via random deletion, and embeds an auxiliary bipartite graph encoding the forbidden complex with admissible cycles (Keevash et al., 2020). Dense link analysis guarantees a homeomorph under sufficient density.

For CW-complex realizations, disk-coverability criteria are introduced: induced 4-cycles in the 1-skeleton shadow of a host 3-graph must bound "disk" subcomplexes. Probabilistic partitioning then ensures those boundaries are realized disjointly (Sankar, 16 Aug 2024).

All constructions take care to balance the density of bad structures—admissible cycles, high-degree faces, etc.—via probabilistic arguments, multiplicity counting, and sharp extremal inequalities (such as path-counting for disk failure).

4. Generalized and Spectral Turán-Type Bounds

Connections with uniform hypergraph Turán numbers are universal: for a $(k-1)$-dimensional complex $F$, the extremal number satisfies

$$\operatorname{ex}_k^c(n,\mathcal{F}) \leq \operatorname{ex}(n,F) \leq \operatorname{ex}_k^{c+}(n,\mathcal{F}) + \sum_{r=0}^{k-1} \binom{n}{r}$$

where $\mathcal{F}$ is the family whose downward-closure contains $F$ (Axenovich et al., 18 Aug 2025). For many classes—e.g., trivial complexes, matchings, cycles—these bounds are tight up to error $o(n^{k-1})$. In general, extremal behavior may be much larger or smaller due to layer incompleteness and combinatorial jumps.

Spectral approaches, using signless Laplacians and Betti numbers, provide additional upper bounds. For pure $r$-dimensional complexes with no $r$-holes ($\beta_r = 0$), the spectral radius satisfies

$q_{r-1}(K) \leq r n - r^2 + 1$

with equality precisely for the "tented complex" $T_n^r$ (Fan et al., 30 Jul 2025). Upper bounds on Betti numbers yield parallel extremal bounds, and the spectral methods can recover leading order exponents for spherical Turán-type problems, albeit typically with weaker constants.

5. Representative Constructions and Explicit Examples

Extremal examples for these bounds include:

• Tented complex $T_n^r$: maximal for spectral radius with no $r$-holes.
• Perfect matching split construction: for two-layer incomplete complexes, partitions $V$ and spreads maximal size across dimensions (Conlon et al., 2023).
• Complete up to dimension $k-1$ plus an $F$-free $k$-graph: yields tight asymptotic bound for certain uniform forbidden complexes.

For 2-complexes generated by a small number of triples, non-trivial behavior is exhibited: e.g., complexes $F_1$, $F_2$ have extremal number $n^3/27 + o(n^3)$, not matching the trivial lower bound (Axenovich et al., 18 Aug 2025). These examples reveal new phenomena such as incompleteness and layer interaction unprecedented in classical hypergraph settings.

6. Open Questions, Conjectures, and Generalizations

Major open problems include:

• Determining for which complexes $F$ the trivial lower bound $$\operatorname{ex}(n,F) \geq \operatorname{ex}_k^{c+}(n,F^k) + \sum_{r=0}^{k-1} \binom{n}{r}$$ is tight (Axenovich et al., 18 Aug 2025, Conlon et al., 2023).
• Establishing sharp exponents for sphere Turán problems in all dimensions, potentially matching the conditional exponent $\lambda_d = (d+1)/(2^{d+1} - 2)$ (Newman et al., 8 Mar 2024).
• Characterizing all real exponents arising as growth rates $\Theta(n^t)$ in simplicial extremal functions; all integers $\geq k-1$ are known to occur, while the rational case is open.
• Extending Turán-type bounds to joint constraints on multiple Betti numbers, non-pure complexes, or other topological invariants (Fan et al., 30 Jul 2025).
• Improving universal exponents, particularly by enhancing bipartite Turán bounds for shadow graphs in CW-realizations (Sankar, 16 Aug 2024).
• Exploring signless $p$-Laplacian approaches and normalized Laplacian extremals.

The emerging synthesis of topological, spectral, and probabilistic arguments defines a new horizon in combinatorial extremal theory. Techniques are rapidly advancing—in particular, by identifying structural symmetries, leveraging density increments, and introducing concepts such as disk-coverability, new tight bounds and universal laws are within reach for Turán-type problems in simplicial complexes.