![](/files/happy5.png)
Generalised Empty Hexagon Conjecture
![$ \ell\geq3 $](/files/tex/06509000d5e4893e990a6bf4e2deca4af4e82a6c.png)
![$ f(\ell) $](/files/tex/4d2a38aa4584bb0aecf3b85c5e0fc3f83263108c.png)
![$ f(\ell) $](/files/tex/4d2a38aa4584bb0aecf3b85c5e0fc3f83263108c.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
Here an empty hexagon in a set of points consists of a subset
of six points in convex position with no other point in
in the convex hull of
. The
case of the conjecture (that is, for point sets in general position) was an outstanding open problem for many years, until its solution by Gerken [G] and Nicolas [N]. Valtr [V] found a simple proof.
Bibliography
[G] Tobias Gerken. Empty Convex Hexagons in Planar Point Sets, Discrete Comput Geom (2008) 39:239–272, MathSciNet
[N] Carlos M. Nicolas. The Empty Hexagon Theorem, Discrete Comput Geom 38:389–397 (2007), MathSciNet.
[V] Pavel Valtr, On Empty Hexagons, in: J. E. Goodman, J. Pach, and R. Pollack, Surveys on Discrete and Computational Geometry, Twenty Years Later, Contemp. Math. 453, AMS, 2008, pp. 433-441.
* indicates original appearance(s) of problem.