Simplicial Complexes and Extremal Theory

• Simplicial complexes are non-uniform, downward closed hypergraphs that include all subsets of each edge, thereby generalizing classical Turán problems.
• They enable the study of extremal numbers by relating maximal uniform layers to Turán numbers and distinguishing between trivial and nontrivial cases.
• Explicit counterexamples reveal jump phenomena where slight structural changes lead to higher edge counts than predicted by trivial bounds.

A simplicial complex is a non-uniform hypergraph whose edge set is downward closed: if an edge is present, all of its subsets are also present. The Turán-type extremal problem for simplicial complexes centers on the extremal number $\mathrm{ex}(n,\mathbf{F})$: the maximal possible number of edges in an $n$-vertex simplicial complex that avoids a fixed forbidden simplicial complex $\mathbf{F}$. This notion extends the classical Turán problems from uniform graphs and hypergraphs to the non-uniform, downward-closed setting. Recent work has elucidated the close relationship between these extremal numbers and generalized Turán numbers for hypergraphs, identified large classes of "trivial" complexes (in terms of extremal behavior), and exhibited explicit examples where standard bounds fail to hold.

1. Abstract Simplicial Complexes and Extremal Functions

An abstract simplicial complex $\mathbf{F}=(V,E)$ is a non-uniform hypergraph—its edges are finite subsets of $V$—with two essential properties:

• Downward closure: If $e\in E$ and $f\subseteq e$, then $f\in E$.
• Containment of singletons (and the empty set): All vertices, and even the empty set, are in $E$; in particular, $\mathbf{F}$ has no isolated vertices.

The dimension of $\mathbf{F}$ is $\max\{|e| : e \in E\} - 1$.

Given a fixed complex $\mathbf{F}$, the extremal number $\mathrm{ex}(n, \mathbf{F})$ is the maximal possible number of edges in a simplicial complex on $n$ vertices that contains no copy of $\mathbf{F}$—that is, avoids all injective maps $V(\mathbf{F})\to [n]$ which send $E(\mathbf{F})$ into $E(\mathcal{C})$.

2. Turán Numbers and Their Generalizations

Turán-type extremal problems in classical graph and hypergraph settings focus on maximizing edge counts in $F$-free $k$-graphs. In the simplicial complex setting, the problem is intricately linked to generalized Turán numbers of uniform hypergraph layers of the complex.

For a non-uniform complex $\mathbf{F}$ with largest edge size $k$, denote its $k$-uniform layer by $\mathbf{F}^k$. The following lower bound emerges: $$\mathrm{ex}(n, \mathbf{F})\ \geq\ e_k^{(c+)}(n, \mathbf{F}^k)\ +\ \sum_{r=0}^{k-1} \binom{n}{r}$$ where $e_k^{(c+)}(n, \mathbf{F}^k)$ is the maximum number of $r$-cliques with $r\geq k$ in a $k$-uniform $\mathbf{F}^k$-free hypergraph. The summation term accounts for all lower-dimensional faces, which always exist by downward closure. This framework allows one to relate extremal functions in the non-uniform (simplicial) world to uniform Turán numbers.

3. Classes of "Trivial" and Nontrivial Complexes

The paper systematically characterizes large families of simplicial complexes for which the lower bound above is sharp (the complex is called "trivial") or asymptotically sharp ("asymptotically trivial"). Key results include:

• Trivial complexes: If the set of maximal faces $(\mathbf{F})$ coincides with $E(\mathbf{F}^k)$ or consists of pairwise disjoint faces, then

$$\mathrm{ex}(n, \mathbf{F}) = e_k^{(c+)}(n, \mathbf{F}^k) + \sum_{r=0}^{k-1} \binom{n}{r}$$

holds exactly for all large $n$. Also, if $|\mathbf{F}| \leq 2$, the result is trivial by direct enumeration.

• Asymptotically trivial complexes: For families of complexes containing a $k$-matching $M_t^k$ and contained in the downward closure of a linear cycle $C_{2t}^k$, the extremal number is asymptotically determined by the lower bound.

These results generalize and extend the work of Conlon, Piga, and Schülke on hypergraph extremal numbers and provide a taxonomy for the extremal behavior of downward-closed complexes (Axenovich et al., 18 Aug 2025).

4. Counterexamples: Jump Phenomena Beyond the Trivial Bound

Not all simplicial complexes behave trivially. The authors construct explicit 2-dimensional examples (labelled $F_1$, $F_2$, $F_3$, $F_4$) with extremal numbers that overshoot the trivial bound:

Example	Trivial Bound	Actual Asymptotics
$F_1$, $F_2$	$O(n^2)$	$\frac{n^3}{27} + o(n^3)$
$F_3$	$O(n^2)$	$\Theta(n^3)$
$F_4$	$O(n^2)$	$\Omega(n^{5/2}),\ O(n^{15/4})$

For $F_1$ and $F_2$, although the forbidden configurations are minimal (a single $3$-edge or a size-$2$ matching), $\mathrm{ex}(n,F_i)$ grows cubicly rather than quadratically. These examples demonstrate that slight modifications in the structure or choice of maximal faces can induce discontinuous “jump” phenomena in the extremal number, breaking the prediction of the trivial bound (Axenovich et al., 18 Aug 2025).

5. Connections with Generalized and Berge Turán Problems

These findings reinforce the deep connection between extremal theory for simplicial complexes and generalized Turán problems in hypergraphs. For instance, consider $\mathbf{F}$ whose maximal $k$-faces form $K_{t+1}^k$. Then for $n\gg t\geq k\geq 2$,

$$\mathrm{ex}(n, \mathbf{F}) = (1+o(1)) e_k(n, K_t^k, K_{t+1}^k)$$

where $e_k(n, K_t^k, K_{t+1}^k)$ is the generalized Turán number counting copies of $K_t^k$ in $K_{t+1}^k$-free hypergraphs. The work also relates to Berge hypergraphs: every simplicial complex (by downward closure) gives rise to a Berge-$\mathbf{F}$-free hypergraph when forbidding a copy of $\mathbf{F}$. Thus, the extremal results have direct implications for longstanding open problems and constructions in the theory of Berge hypergraphs.

6. Open Problems and Future Directions

Classification of trivial and nontrivial complexes is incomplete. Natural questions posed include:

• For which infinite families is the trivial bound sharp?
• For which uniformity layers (i.e., layers of dimension $s$ between $2$ and $k$) is the lower bound sharp?
• For a fixed simplicial complex, what exponents $t$ (with $\mathrm{ex}(n,\mathbf{F})=\Theta(n^t)$) are feasible, paralleling the open questions for uniform hypergraph Turán problems?

The interplay between combinatorial structure of maximal faces, uniformity layers, and extremal function growth exponents remains an open field for further research, as does the extension of these techniques to other combinatorial invariants of complexes and to applications involving Berge hypergraphs.

7. Summary of Key Formulas and Relations

• Lower bound on extremal number:

$$\mathrm{ex}(n, \mathbf{F})\ \geq\ e_k^{(c+)}(n, \mathbf{F}^k)\ +\ \sum_{r=0}^{k-1} \binom{n}{r}$$

• For maximal faces forming $K_{t+1}^k$:

$$\mathrm{ex}(n, \mathbf{F}) = (1 + o(1))\, e_k(n, K_t^k, K_{t+1}^k)$$

These relationships exemplify how clique densities in the maximal uniform layer dominate the overall extremal count, modulo lower-dimensional contributions, except in nontrivial configurations where additional combinatorial constraints sharply increase the extremal number (Axenovich et al., 18 Aug 2025).

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In summary, Turán problems for simplicial complexes not only generalize foundational questions from graph and hypergraph theory to downward-closed set systems, but also reveal rich new stratifications and exceptional phenomena that reflect the underlying structure of maximal faces. The latest research both clarifies large zones of "trivial" behavior and establishes sharp counterexamples, motivating further investigations into the extremal combinatorics of non-uniform hypergraphs and their connections to classical and Berge hypergraph theory.