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Finding Sets of Points without Empty Convex 6-Gons

Abstract. Erdös asked whether every large enough set of points in general position in the plane contains six points that form a convex 6-gon without any points from the set in its interior. In this note we show how a set of 29 points was found that contains no empty convex 6-gon. To this end a fast incremental algorithm for finding such 6-gons was designed and implemented and a heuristic search approach was used to find promising sets. The experiments also led to two observations that might be useful in proving that large sets always contain an empty convex 6-gon.

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Mark Overmars, . Finding Sets of Points without Empty Convex 6-Gons . Discrete Comput Geom 29, 153–158 (2002).

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  • General Position
  • Heuristic Search
  • Search Approach
  • Incremental Algorithm
  • Empty Convex