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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 $E$?

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

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  • $\begingroup$ You are right. I was actually looking for non-trivial ones, but the question is not well put. $\endgroup$ – Alfred Jun 1 at 12:00

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