# Configuration of high-dimensional spheres

Let $$S = \{S_1, \dots ,S_n\}$$ be a finite set of $$d$$-dimensional spheres, and let $$E$$ be a combination of intersections between them, where an intersection is a rule of the form $$S_i \cap S_j \subset S_k$$. Given any number of spheres and any combination of intersections, is it always possible to find a configuration of spheres embedded in $$\mathbb R^d$$ which satisfies all and only the intersections in $$E$$? Hence, this configuration must not contain any intersection that is not present in $$E$$.

Side question:

If the answer is negative, but dependent on the dimension, is the following true: Given any number of spheres and any combination of intersections, there exist a finite dimension d such that is it always possible to find a configuration of d-dimensional spheres embedded in $$\mathbb R^d$$ which satisfies all and only the intersections in $$E$$?

• Well, it's clearly not possible for $d=0$ and $d=1$, for a start. – joriki May 31 at 23:46

No, because some rules of your form imply others. For example, if you have $$S_1\cap S_2\subseteq S_3$$ and $$S_2\cap S_3\subseteq S_4$$, then you automatically also have $$S_1\cap S_2\subseteq S_4$$.
Notice that this doesn't depend on the $$S_i$$'s being spheres; it's true for any sets. There are lots more trivial implications like this, plus probably some nontrivial ones that do depend on these sets being spheres.