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I remembered a question I once heard and never found the answer, maybe you can help me with that:

given the set $S$. $S$ is a set of points on $\Bbb R^n, n\ge2$, which is finite and has more than $2$ points in it. prove or disprove the following statement:

it is possible to create a line that goes through exactly $2$ of the points iff it is impossible to create a line that goes through all of the points in the set.

at start I tried to show a counterexample. but I couldn't think about any.

after that i tried to show that it is true, I started by simplifying the question for $\Bbb R^2$. it is obvious that if the points in the set are over the same line it is impossible to create a line that goes through only $2$ of them so i'll not add to here the proof for this. Now i dont know how to continue, how can i show that for any other case there is a line that goes through only $2$ of the points?

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ℋolo
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1 Answers1

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If $n=2$, then this is the Sylvester–Gallai theorem.

José Carlos Santos
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  • As for $n>2$, can I say that every set of $3$ or more points construct a plane of $\Bbb R^2$ and then because I showed of sylvester-gallai theorem it is also true for all $n\ge 2$ – ℋolo Oct 23 '17 at 18:53
  • @Holo Delete the “of $\mathbb{R}^2$” part and I'll agree. – José Carlos Santos Oct 23 '17 at 22:35