Computational geometry column 50
ACM SIGACT News
Computational Geometry: Theory and Applications
Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons
SIAM Journal on Discrete Mathematics
Monotonic polygons and paths in weighted point sets
CGGA'10 Proceedings of the 9th international conference on Computational Geometry, Graphs and Applications
Computational geometry column 53
ACM SIGACT News
Large convex holes in random point sets
Computational Geometry: Theory and Applications
Monochromatic empty triangles in two-colored point sets
Discrete Applied Mathematics
On planar point sets with the pentagon property
Proceedings of the twenty-ninth annual symposium on Computational geometry
On the Erdős-Szekeres n-interior-point problem
European Journal of Combinatorics
Unsolved problems in visibility graphs of points, segments, and polygons
ACM Computing Surveys (CSUR)
Lower bounds for the number of small convex k-holes
Computational Geometry: Theory and Applications
Computational Geometry: Theory and Applications
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Let P be a finite set of points in general position in the plane. Let C(P) be the convex hull of P and let CiP be the ith convex layer of P. A minimal convex set S of P is a convex subset of P such that every convex set of P ∩ C(S) different from S has cardinality strictly less than |S|. Our main theorem states that P contains an empty convex hexagon if C1P is minimal and C4P is not empty. Combined with the Erdos-Szekeres theorem, this result implies that every set P with sufficiently many points contains an empty convex hexagon, giving an affirmative answer to a question posed by Erdos in 1977.