The arity of a (non-empty) convexity space $ \left(X, \mathscr{C} \right) $
is defined as follow: $$ 0 \lt ar\left(\mathscr{C}\right) \le n \iff \left( C \in \mathscr{C} \iff\ C = \bigcup \{co\left(F\right) \vert F \subseteq C;\ card(F) \le n \} \right)$$
where $n \in \omega$ and $co$ denotes the hull operator. We take the least such $n$ to be the arity.
Furthermore, we say that $ar\left(\mathscr{C}\right) = \infty$ if $\mathscr{C}$ is not of arity $\le n$ for any $n$.
Finding natural example of spaces of arity one and two is easy.
For arity $2$ just think of metric spaces with standard convexity.
For arity $1$ take $X = \omega$ and elements of $\mathscr{C}$ to be sets of the form $U(k) = \{ n \in \omega \vert n \ge k \}$ and the empty set.
There are also spaces of arity $3$ that arise naturally.
Taking $X$ the underlyng set of a group and $\mathscr{C}$ the set of left-cosets of subgroups (together with the empty set). These convexities have arity $\le 3$ (Jamison [1974]) (but not always $=3$, take $\mathbb{Z}$ for example)
You can have space of arity $n$ for any $n \in \omega$. You can also have infinite arity.
Take $X_n = \omega$ and $\mathscr{C}_n = \{F \subseteq \omega \ \vert \ card(F) \le n \} \cup \{ X_n \}$ cleary, it has arity $n+1$
Then take $(X,\mathscr{C}) = \coprod_{n \in \omega} (X_n,\mathscr{C}_n) $. It has infinite arity.
(1) Are there any convexity spaces of infinite arity that arise naturally ?
(2) Are there examples convexity spaces of arbitrarily large (integer) arity (or just $\gt 3$) that arise naturally?
By "naturally", I mean that those spaces weren't constructed for the sole sake of having arity $n$.
