On the minimum size of a point set containing a 5-hole and a disjoint 4-hole
2011 ◽
Vol 48
(4)
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pp. 445-457
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Keyword(s):
Let H(k; l), k ≦ l denote the smallest integer such that any set of H(k; l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117–122, 2004] proved that 12 ≦ H(4, 5) ≦ 14. Very recently, using a Ramseytype result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481–507, 2007], Hosono and Urabe [Kyoto CGGT, LNCS 4535, 90–100, 2008] improve the upper bound to 13. In this paper, with the help of the same Ramsey-type result, we prove that H(4; 5) = 12.
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2013 ◽
Vol 50
(3)
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pp. 331-354
Keyword(s):
2006 ◽
Vol 113
(3)
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pp. 385-419
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2019 ◽
Vol 11
(01)
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pp. 1950004
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