TY - JOUR
AU - Tao, Terence
AB - Let 
$$P$$


P


 be a set of 
$$n$$


n


 points in the plane, not all on a line. We show that if 
$$n$$


n


 is large then there are at least 
$$n/2$$



n
/
2




ordinary lines, that is to say lines passing through exactly two points of 
$$P$$


P


. This confirms, for large 
$$n$$


n


, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than 
$$n-C$$



n
-
C



 ordinary lines for some absolute constant 
$$C$$


C


. We also solve, for large 
$$n$$


n


, the “orchard-planting problem”, which asks for the maximum number of lines through exactly 3 points of 
$$P$$


P


. Underlying these results is a structure theorem which states that if 
$$P$$


P


 has at most 
$$Kn$$



K
n



 ordinary lines then all but O(K) points of 
$$P$$


P


 lie on a cubic curve, if 
$$n$$


n


 is sufficiently large depending on 
$$K$$


K


.
TI - On Sets Defining Few Ordinary Lines
JO - Discrete & Computational Geometry
DO - 10.1007/s00454-013-9518-9
DA - 2013-06-27
UR - https://www.deepdyve.com/lp/springer-journals/on-sets-defining-few-ordinary-lines-84RKHSdj5V
SP - 409
EP - 468
VL - 50
IS - 2
DP - DeepDyve
ER -