From graph theory:

Embed a fully-connected graph without multiple edges into a 2-dimensional euclidean space (the plane), some edges will intersect each-other if the number of vertices is at least a small constant (6), however, the edges will not intersect each-other for any number of vertices if the same graph is embedded into an euclidean space of dimension n >= 3.

The is a weaker version of the above statement: Embed a graph without multiple edges with minimum degree of 4 into a 2-dimensional euclidean space, some edges will intersect each-other if the number of vertices is at least a small constant (5), however, the edges will not intersect each-other if the same graph is embedded into an euclidean space of dimension n >= 3.

There are similar results for spherical, cylindrical, toroidal and other topological spaces.

From graph theory:

Embed a fully-connected graph without multiple edges into a 2-dimensional euclidean space (the plane), some edges will intersect each-other if the number of vertices is at least a small constant (6), however, the edges will not intersect each-other for any number of vertices if the same graph is embedded into an euclidean space of dimension $n \geq 3$.

There is a weaker version of the above statement: Embed an arbitrary graph without multiple edges with minimum degree of 4 into a 2-dimensional euclidean space, some edges will intersect each-other if the number of vertices is at least a small constant (5), however, the edges will not intersect each-other if the same graph is embedded into an euclidean space of dimension $n \geq 3$.

There are similar results for spherical, cylindrical, toroidal and other topological spaces.