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Let $S = \{S_1, \dots ,S_n\}$ be a finite set of $d$-dimensional spheres with the same radius, 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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    $\begingroup$ The problem is similar to that of specifying for each $i,j$ whether or not $S_i$ is inside $S_j$, where now the spheres need not have the same radius. Felsner, Trotter, and Fishburn showed that some specifications could not be achieved in any dimension. See page.math.tu-berlin.de/~felsner/Paper/sphere.pdf. $\endgroup$
    – Richard Stanley
    Commented Jun 1, 2020 at 0:19

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If I am not misunderstanding, it could be that this $\mathbb{R}^2$ example shows that $S_5$ cannot simultaneously satisfy these relationships: \begin{eqnarray} S_1 \cap S_2 & \subset & S_5 \\ S_3 \cap S_4 & \subset & S_5 \\ S_2 \cap S_3 & \not\subset & S_5 \\ S_1 \cap S_4 & \not\subset & S_5 \\ \end{eqnarray}

          Circ5
          Here four circles have same radii, but in general the radii may differ.
This is related to OEIS A250001. I recall that Jonathan Wild proved the impossibility of the above configuration, but I have no reference. This example may not resolve your question, because I am uncertain if Wild's conditions on drawing circles are exactly equivalent to your intersection relations.

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  • $\begingroup$ It actually looks relevant, thanks! Are you aware of any multi-dimensional extension of this? $\endgroup$
    – Alfred
    Commented May 31, 2020 at 22:46
  • $\begingroup$ @Alfred: I am unaware of extensions to $\mathbb{R}^d$ for $d \ge 3$. It is already quite difficult for $\mathbb{R}^2$: $16951$ ways to draw $5$ circles! $\endgroup$
    – Joseph O'Rourke
    Commented May 31, 2020 at 22:51

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