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A pseudo-disk arrangement is a collection of planar bodies whose boundaries are Jordan curves that pairwise intersect at most twice. I would like to know if given seven points in the plane whether it is possible to find a pseudo-disk arrangement with $\binom 73$ disks, such that for each triple of the seven points there is one disk containing exactly them from the set of seven points. I'm also interested in what kind of theorems there are to conclude the non-existence of such embeddings.

I've just realized that I don't even know the answer if we replace seven with five. For five points we can find a pseudo-ball arrangement is $3d$; take a tetrahedron with a point inside, and slightly perturb the ten triangles they span. Can we do more points in $3d$? In $4d$ any number of points are OK if we aim for triples (just like pairs would be OK in $3d$).

A pseudo-disk arrangement is a collection of planar bodies whose boundaries are Jordan curves that pairwise intersect at most twice. I would like to know if given seven points in the plane whether it is possible to find a pseudo-disk arrangement with $\binom 73$ disks, such that for each triple of the seven points there is one disk containing exactly them from the set of seven points. I'm also interested in what kind of theorems there are to conclude the non-existence of such embeddings.

I've just realized that I don't even know the answer if we replace seven with five. For five points we can find a pseudo-ball arrangement is $3d$; take a tetrahedron with a point inside, and slightly perturb the ten triangles they span. Can we do more points in $3d$? In $4d$ any number of points are OK if we aim for triples (just like pairs would be OK in $3d$).

Please note that I've updated the question a couple of times to make the problem more clear, some of the comments and Jeff's answer are for older versions.

A pseudo-disk arrangement is a collection of planar bodies whose boundaries are Jordan curves that pairwise intersect at most twice. I would like to know if given seven points in the plane whether it is possible to find a pseudo-disk arrangement with $\binom 73$ disks, such that for each triple of the seven points there is one disk containing exactly them from the set of seven points. I'm also interested in what kind of theorems there are to conclude the non-existence of such embeddings.

I've just realized that I don't even know the answer if we replace seven with five.

A pseudo-disk arrangement is a collection of planar bodies whose boundaries are Jordan curves that pairwise intersect at most twice. I would like to know if given seven points in the plane whether it is possible to find a pseudo-disk arrangement with $\binom 73$ disks, such that for each triple of the seven points there is one disk containing exactly them from the set of seven points. I'm also interested in what kind of theorems there are to conclude the non-existence of such embeddings.

I've just realized that I don't even know the answer if we replace seven with five. For five points we can find a pseudo-ball arrangement is $3d$; take a tetrahedron with a point inside, and slightly perturb the ten triangles they span. Can we do more points in $3d$? In $4d$ any number of points are OK if we aim for triples (just like pairs would be OK in $3d$).

We say that two connected planar bodies cross if their union minus their intersection has more than two connected components. (Here we suppose that their boundaries are nice and intersect in finitely many points. For example, we can suppose that each body is a polygon and their vertices are in general position.) If no two member of a collection of bodies cross, then this is also called a pseudo-disk arrangement. I would like to know if given seven points in the plane whether it is possible to find $\binom 73$ pairwise non-crossing connected bodies, such that for each triple of the seven points there is one body containing exactly them from the set of seven points. I'm also interested in what kind of theorems there are to conclude the non-existence of such embeddings.

I've just realized that I don't even know the answer for the 4-simplex, i.e., all triples of five points.

A pseudo-disk arrangement is a collection of planar bodies whose boundaries are Jordan curves that pairwise intersect at most twice. I would like to know if given seven points in the plane whether it is possible to find a pseudo-disk arrangement with $\binom 73$ disks, such that for each triple of the seven points there is one disk containing exactly them from the set of seven points. I'm also interested in what kind of theorems there are to conclude the non-existence of such embeddings.

I've just realized that I don't even know the answer if we replace seven with five.