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Question:

what is known about the sequence $\mathbb{X}\subset \mathbb{N}_0$ such that for each $k\in \mathbb{X}$ there exists a set of $n$ points in general position in the Euclidean plane such that the set of line segments that connect pairs of the points intersect in exactly $k$ inner points?

In general position in this context means that no three are collinear.

I don't want to rule out the case where 3 lines intersect in an point, so I ask to indicate in answers to which case they relate, to stable ones, where the number of intersections doesn't change if the point locations are modified by a sufficiently small distance for all directions, or to the critical on where we have a direciton, in which a point can't be moved without changing the number o intersections, no matter how small the positive distance is.

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  • $\begingroup$ Can 3 line segments intersect in one point? $\endgroup$
    – Max Alekseyev
    Commented Jun 8, 2023 at 3:03
  • $\begingroup$ @MaxAlekseyev I am tempted to allow both variants,meaning that both cases are interesting in their own right. $\endgroup$
    – Manfred Weis
    Commented Jun 8, 2023 at 5:25
  • $\begingroup$ It's worth to compute first few terms and check if they are present in oeis.org $\endgroup$
    – Max Alekseyev
    Commented Jun 8, 2023 at 12:25
  • $\begingroup$ @MaxAlekseyev my guess would be that no solution exists for 2 intersections $\endgroup$
    – Manfred Weis
    Commented Jun 8, 2023 at 13:41
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    $\begingroup$ If $n$ points were the vertices of a convex polygon, and no 3 line segments were allowed to intersect at one point, then the answer would be $\binom{n}{4}$. $\endgroup$
    – Max Alekseyev
    Commented Jun 9, 2023 at 21:31

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