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Assume you have a set X (of points) and subsets $A_i$ with the following conditions:

(1): For any two points in X exactly one of the sets contain them. (2): Any two subsets intersect at most at one point.

The question is: Can we find $|X|$ points in the plane such that the $A_i$ are exactly the lines through them.

Possible combinatorial obstructions are Sylvester's theorem, Desarges and Papus Theorem as indicated below. What other obstructions are there? Can these graphs be characterized?

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Note that the number of ordinary lines is actually much bigger than one. So if you have a design with few subsets of size $2,$ it cannot be realized.

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Of course no. For example, Pappus and Desargues theorems are obstructions.

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  • $\begingroup$ It is not immediate that the three conditions the OP states do not imply Pappus/Desargues (I am not saying that this is surprising, merely that it is not immediarte). $\endgroup$
    – Igor Rivin
    Commented Sep 3, 2017 at 20:21
  • $\begingroup$ I've edited the question accordingly. $\endgroup$
    – Espace' etale
    Commented Jan 17, 2018 at 22:34

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