Editing Heilbronn triangle problem (section)

==Upper bounds==
Every set of <math>n</math> points in the unit square forms a triangle of area at most inversely proportional {{nowrap|to <math>n</math>.}} One way to see this is to [[Point set triangulation|triangulate]] the [[convex hull]] of the given point {{nowrap|set <math>S</math>,}} and choose the smallest of the triangles in the triangulation. Another is to sort the points by their {{nowrap|<math>x</math>-coordinates,}} and to choose the three consecutive points in this ordering whose {{nowrap|<math>x</math>-coordinates}} are the closest together. In the first paper published on the Heilbronn triangle problem, in 1951, [[Klaus Roth]] proved a stronger upper bound {{nowrap|on <math>\Delta(n)</math>,}} of the form{{r|roth}}
<math display=block>\Delta(n)=O\left(\frac{1}{n\sqrt{\log\log n}}\right).</math>
The best bound known to date is of the form
<math display=block>\Delta(n)\leq\frac{\exp{\left(c\sqrt{\log n}\right)}}{n^{8/7}},</math>
for some {{nowrap|constant <math>c</math>,}} proven by {{harvtxt|Komlós|Pintz|Szemerédi|1981}}.{{r|kps81}}

A new upper bound equal to <math>n^{-\frac{8}{7}-\frac{1}{2000}}</math> was proven by {{harvtxt|Cohen|Pohoata|Zakharov|2023}}.{{r|cpz23|sloman}}