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Is Sylvester's theorem the only obstruction Obstruction to embedding a point-line graph in $R^2$

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. (3): One of the subsets is of size 2 exactly.

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

(3) is exactly Possible combinatorial obstructions are Sylvester's theorem, I am wondering if that is the only obstructionDesarges and Papus Theorem as indicated below. What other obstructions are there? Can these graphs be characterized?

Is Sylvester's theorem the only obstruction to embedding a point-line graph in $R^2$

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. (3): One of the subsets is of size 2 exactly.

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

(3) is exactly Sylvester's theorem, I am wondering if that is the only obstruction.

Obstruction to embedding a point-line graph in $R^2$

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?

Source Link
Espace' etale
Espace' etale
  • 478
  • 4
  • 9

Is Sylvester's theorem the only obstruction to embedding a point-line graph in $R^2$

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. (3): One of the subsets is of size 2 exactly.

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

(3) is exactly Sylvester's theorem, I am wondering if that is the only obstruction.