Dimension of complex on 2-sphere

Question

Given a 2-sphere $S^2$ and a set of points $A$ embedded on its surface, if we were to consider a pair of points $(a,b) \in A$, and construct each point's $k$-nearest neighbor set, say $K(a)$ and $K(b)$, respectively. We add an edge or arc (along the surface of the sphere) between $a$ and $b$, if $K(a) \cap K(b) \neq \emptyset$ (non-empty intersection) and do this for all possible pairs in $A$. If we were to then construct a simplicial complex $X$ from such a graph defining the cliques (complete subgraphs) as simplices, can $dim(X) >2$? I ask because I imagine that at best I can only arrive at $2$-simplices (triangles), as these edges can overlap for a $4$-clique?