منابع مشابه
Empty Monochromatic Simplices
Let S be a k-colored (finite) set of n points in R, d ≥ 3, in general position, that is, no (d+1) points of S lie in a common (d−1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3 ≤ k ≤ d we provide a lower bound of Ω(nd−k+1+...
متن کاملMonochromatic simplices of any volume
We give a very short proof of the following result of Graham from 1980: For any finite coloring of R, d ≥ 2, and for any α > 0, there is a monochromatic (d + 1)tuple that spans a simplex of volume α. Our proof also yields new estimates on the number A = A(r) defined as the minimum positive value A such that, in any rcoloring of the grid points Z of the plane, there is a monochromatic triangle o...
متن کاملEmpty Monochromatic Triangles
We consider a variation of a problem stated by Erdös and Guy in 1973 about the number of convex k-gons determined by any set S of n points in the plane. In our setting the points of S are colored and we say that a spanned polygon is monochromatic if all its points are colored with the same color. As a main result we show that any bi-colored set of n points in R in general position determines a ...
متن کاملAn Introduction to Empty Lattice Simplices
We study simplices whose vertices lie on a lattice and have no other lattice points. Suchèmpty lattice simplices' come up in the theory of integer programming, and in some combi-natorial problems. They have been investigated in various contexts and under varying terminology Can thèemptiness' of lattice simplices bèwell-characterized' ? Is theirìattice-width' small ? Do the integer points of the...
متن کاملOn Empty Lattice Simplices in Dimension 4
We give an almost complete classification of empty lattice simplices in dimension 4 using the conjectural results of Mori-Morrison-Morrison, later proved by Sankaran and Bober. In particular, all of these simplices correspond to cyclic quotient singularities, and all but finitely many of them have width bounded by 2.
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2014
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-013-9565-2