Large convex holes in random point sets

O. Aichholzer, [Empty] [colored] k-gons. Recent results on some Erdős–Szekeres type problems, in: Proc. XIII Encuentros de Geometría Computacional, Zaragoza, Spain, 2009, pp. 43–52. Alon, 2008 Bárány, 1987, Empty simplices in Euclidean space, Canad. Math. Bull., 30, 436, 10.4153/CMB-1987-064-1 Bárány, 2009, Longest convex chains, Random Structures Algorithms, 35, 137, 10.1002/rsa.20269 Bárány, 2001, A note on Sylvesterʼs four-point problem, Studia Sci. Math. Hungar., 38, 73 Bárány, 1999, Sylvesterʼs question: the probability that n points are in convex position, Ann. Probab., 27, 2020, 10.1214/aop/1022677559 Bárány, 2004, Planar point sets with a small number of empty convex polygons, Studia Sci. Math. Hungar., 41, 243 Buchta, 2006, The exact distribution of the number of vertices of a random convex chain, Mathematika, 53, 247, 10.1112/S0025579300000127 Brass, 2005 Erdős, 1935, A combinatorial problem in geometry, Compositio Math., 2, 463 Erdős, 1978, Some more problems on elementary geometry, Austral. Math. Soc. Gaz., 5, 52 Gerken, 2008, Empty convex hexagons in planar point sets, Discrete Comput. Geom., 39, 239, 10.1007/s00454-007-9018-x Harborth, 1978, Konvexe Fünfecke in ebenen Punktmengen, Elem. Math., 33, 116 Horton, 1983, Sets with no empty convex 7-gons, Canad. Math. Bull., 26, 482, 10.4153/CMB-1983-077-8 Koshelev, 2007, The Erdős–Szekeres problem, Dokl. Math., 76, 603, 10.1134/S106456240704031X Morris, 2000, The Erdős–Szekeres problem on points in convex position — a survey, Bull. Amer. Math. Soc., 37, 437, 10.1090/S0273-0979-00-00877-6 Nicolás, 2007, The empty hexagon theorem, Discrete Comput. Geom., 38, 389, 10.1007/s00454-007-1343-6 Overmars, 2003, Finding sets of points without empty convex 6-gons, Discrete Comput. Geom., 29, 153, 10.1007/s00454-002-2829-x Rényi, 1963, Über die konvexe Hülle von n zufällig gewählten Punkten, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 2, 75, 10.1007/BF00535300 Rényi, 1964, Über die konvexe Hülle von n zufällig gewählten Punkten, II, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3, 138, 10.1007/BF00535973 J.J. Sylvester, Question 1491, in: The Educational Times, London, April 1864. Valtr, 1995, Probability that n random points are in convex position, Discrete Comput. Geom., 13, 637, 10.1007/BF02574070 Valtr, 1996, The probability that n random points in a triangle are in convex position, Combinatorica, 16, 567, 10.1007/BF01271274 Valtr, 2008, On empty hexagons, vol. 453, 433