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An Erd\"os-Szekeres-typeErdős-Szekeres-type question

Does there exist an integer $N$ such that every set of $\geq N$ points in $\mathbb R^4$ contains six distinct points which are vertices of two intersecting triangles?

More generally, given dimensions $d_1,\dots,d_k$ such that generic affine subspaces of $\mathbb R^d$ of dimensions $d_1,\dots,d_k$ intersect in a point, does every large enough (but finite) set of $\mathbb R^d$ contain $k+\sum_i d_i$ distinct points defining $k$ simplices of dimension $d_1,\dots,d_k$ intersecting in a point?

There are many variations on this problem. For example: Given an integer $k$, does every large enough subset of $\mathbb R^4$ contain the vertices of $k$ triangles (with all $3k$ vertices distinct) such that all triangles intersect pairwise? The corresponding planar problem has a positive answer: By the Erd\"os Erdős Szekeres theorem, every large enough subset of $\mathbb R^2$ (in generic positition) contains $2k$ points in convex position. This defines $k$ "diagonals" which are all pairwise intersecting.

(The Erd\"os-Szekeres Erdős-Szekeres Theorem shows of course that it is enough to consider points in convex position for all questions mentioned above.)

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Does there exist an integer $N$ such that every set of $\geq N$ points in $\mathbb R^4$ contains six distinct points which are vertices of two intersecting triangles?

More generally, given dimensions $d_1,\dots,d_k$ such that generic affine subspaces of $\mathbb R^d$ of dimensions $d_1,\dots,d_k$ intersect in a point, does every large enough (but finite) set of $\mathbb R^d$ contain $k+\sum_i d_i$ distinct points defining $k$ simplices of dimension $d_1,\dots,d_k$ intersecting in a point?

There are many variations on this problem. For example: Given an integer $k$, does every large enough subset of $\mathbb R^4$ contain the vertices of $k$ triangles (with all $3k$ vertices distinct) such that all triangles intersect pairwise? The corresponding planar problem has a positive answer: By the Erd\"os Szekeres theorem, every large enough subset of $\mathbb R^2$ (in generic positition) contains $2k$ points in convex position. This defines $k$ "diagonals" which are all pairwise intersecting.

(The Erd\"os-Szekeres Theorem shows of course that it is enough to consider points in convex position for all questions mentionned mentioned above.)

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An Erd\"os-Szekeres-type question

Does there exist an integer $N$ such that every set of $\geq N$ points in $\mathbb R^4$ contains six distinct points which are vertices of two intersecting triangles?

More generally, given dimensions $d_1,\dots,d_k$ such that generic affine subspaces of $\mathbb R^d$ of dimensions $d_1,\dots,d_k$ intersect in a point, does every large enough (but finite) set of $\mathbb R^d$ contain $k+\sum_i d_i$ distinct points defining $k$ simplices of dimension $d_1,\dots,d_k$ intersecting in a point?

There are many variations on this problem. For example: Given an integer $k$, does every large enough subset of $\mathbb R^4$ contain the vertices of $k$ triangles (with all $3k$ vertices distinct) such that all triangles intersect pairwise? The corresponding planar problem has a positive answer: By the Erd\"os Szekeres theorem, every large enough subset of $\mathbb R^2$ (in generic positition) contains $2k$ points in convex position. This defines $k$ "diagonals" which are all pairwise intersecting.

(The Erd\"os-Szekeres Theorem shows of course that it is enough to consider points in convex position for all questions mentionned above.)