The [Sylvester–Gallai theorem](https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem) in geometry states that, given a finite number of points in the Euclidean plane, either


1. All the points are collinear; or
2. There is a line which contains exactly two of the points.


I think the theorem is also true for circle, plane version and curve version, can you give a proof of these versions?


**Conjecture: (circle version):**



> 
> Given a finite number of points in the Euclidean plane, either
> 
> 
> 1. All the points are concyclic; or
> 2. There is a circle which contains exactly three points.
> 
> 
> 


**Conjecture: (Plane version)**



> 
> Given a finite number of points in the three-dimensional space, either
> 
> 
> 1. All the points in a plane; or
> 2. There is a plane which contains exactly three points.
> 
> 
> 


**Conjecture: (Curve version)**



> 
> Let $n, N$ be the natural number with $(\frac{3n+n^2}{2} < N < \infty)$, give $N$ points in the Euclidean plane, either
> 
> 
> 1. All the points lie on a curve of degree $n$; or
> 2. There is a curve of degree $n$ which contains exactly $\frac{3n+n^2}{2}$ of the points.
> 
> 
> 



Sylvester–Gallai theorem with circle version, plane version and curve version?
The plane version is false as stated. Choose two skew lines $\ell\_1$ and $\ell\_2$ in $\mathbb{R}^3$ and place $\geq 3$ points on each of them. Any three of these points will either all lie on the same $\ell\_i$ (and thus be collinear) or two of them will lie on the same $\ell\_i$, so the plane they span contains that $\ell\_i$ and thus contains three collinear points.


A true version is that there is a plane containing precisely $k$ of the points for which $k-1$ of them are collinear. This is due to [Motzkin](https://doi.org/10.1090/S0002-9947-1951-0041447-9 "zbMATH review at https://zbmath.org/0043.14603") (Trans. AMS, 1951), with a higher dimensional generalization by [Hansen](https://doi.org/10.7146/math.scand.a-10758 "zbMATH review at https://zbmath.org/0156.19501") (Math. Scand. 1965). I took the above counterexample from Hansen's paper.


I think the curve version should be false, but I haven't broken it yet. I have now discovered two papers: [Wiseman and Wilson](https://doi.org/10.1007/BF02187914 "zbMATH review at https://zbmath.org/0643.51009"), (Disc. and Comp. Geom 1988) and [Czapliński et. al.](https://doi.org/10.4171/rsmup/136-13 "zbMATH review at https://zbmath.org/1408.52036"), (Rendiconti del Seminario Matematico della Università di Padova 2016) which pose the curve version as an open question, so it probably isn't easy to resolve.


