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EVERY LARGE POINT SET CONTAINS MANY COLLINEAR POINTS OR AN EMPTY PENTAGON
, 2009
"... We prove the following generalised empty pentagon theorem: for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique” conjecture of Kára, Pór, and Wood ..."
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We prove the following generalised empty pentagon theorem: for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique” conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497–506, 2005].
ON THE GENERAL POSITION SUBSET SELECTION PROBLEM∗
"... Abstract. Let f(n, ) be the maximum integer such that every set of n points in the plane with at most collinear contains a subset of f(n, ) points with no three collinear. First we prove that if O(√n), then f(n, ) Ω(√n / ln ). Second we prove that if O(n(1−)/2), then f(n, ) Ω( n log n), ..."
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Abstract. Let f(n, ) be the maximum integer such that every set of n points in the plane with at most collinear contains a subset of f(n, ) points with no three collinear. First we prove that if O(√n), then f(n, ) Ω(√n / ln ). Second we prove that if O(n(1−)/2), then f(n, ) Ω( n log n), which implies all previously known lower bounds on f(n, ) and improves them when is not fixed. A more general problem is to consider subsets with at most k collinear points in a point set with at most collinear. We also prove analogous results in this setting.
4holes in point sets
, 2012
"... We consider a variant of a question of Erdős on the number of empty kgons (kholes) in a set of n points in the plane, where we allow the kgons to be nonconvex. We show bounds and structural results on maximizing and minimizing the number of general 4holes, and maximizing the number of nonconv ..."
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We consider a variant of a question of Erdős on the number of empty kgons (kholes) in a set of n points in the plane, where we allow the kgons to be nonconvex. We show bounds and structural results on maximizing and minimizing the number of general 4holes, and maximizing the number of nonconvex 4holes. In particular, we show that for n ≥ 9, the maximum number of general 4holes is n