OFFSET
1,11
COMMENTS
The value a(10) = 1 was determined by Harborth, who also constructed a set of 9 points without convex 5-holes. The values a(11) = 2 and a(13) = 3 were determined by Dehnhardt. Aichholzer found point sets showing that a(14) <= 6 and a(15) <= 9, and the exact values a(13) = 3, a(14) = 6, and a(15) = 9 were determined in the Bachelor's thesis of Scheucher, supervised by Aichholzer and Hackl.
The value a(16) = 11 was determined using a ILP/SAT solver. For more information check out the link below with title "On 5-Holes". - Manfred Scheucher, Aug 18 2018
REFERENCES
K. Dehnhardt, Leere konvexe Vielecke in ebenen Punktmengen, PhD thesis, TU Braunschweig, Germany, 1987, in German.
LINKS
O. Aichholzer, M. Balko, T. Hackl, J. Kynčl, I. Parada, M. Scheucher, P. Valtr, and B. Vogtenhuber, A superlinear lower bound on the number of 5-holes, arXiv:1703.05253 [math.CO], 2017.
O. Aichholzer, R. Fabila-Monroy, T. Hackl, C. Huemer, A. Pilz, and B. Vogtenhuber, Lower bounds for the number of small convex k-holes, Computational Geometry: Theory and Applications, 47(5):605-613, 2014.
EuroGIGA - CRP ComPoSe, A set of 13 points with 3 convex 5-holes
EuroGIGA - CRP ComPoSe, A set of 14 points with 6 convex 5-holes
EuroGIGA - CRP ComPoSe, A set of 15 points with 9 convex 5-holes
EuroGIGA - CRP ComPoSe, A set of 16 points with 11 convex 5-holes
H. Harborth, Konvexe Fünfecke in ebenen Punktmengen, Elemente der Mathematik, 33:116-118, 1978, in German.
M. Scheucher, Counting Convex 5-Holes, Bachelor's thesis, Graz University of Technology, Austria, 2013, in German.
M. Scheucher, On 5-Holes.
FORMULA
From Manfred Scheucher, Mar 22 2017: (Start)
a(n) = Omega(n log^(4/5)(n)) and a(n) = O(n^2).
Conjecture: a(n) = Theta(n^2). (End)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Manfred Scheucher, Aug 18 2016
EXTENSIONS
a(16) from Manfred Scheucher, Mar 22 2017
STATUS
approved