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? 



Obstruction to embedding a point-line graph in $R^2$
Note that the number of *ordinary lines* is [actually much bigger than one.](https://terrytao.wordpress.com/tag/dirac-motzkin-conjecture/) So if you have a design with few subsets of size $2,$ it cannot be realized.


