منابع مشابه
Monochromatic 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...
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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 کاملVolume form as volume of infinitesimal simplices
In the context of Synthetic Differential Geometry, we describe the square volume of a “second-infinitesimal simplex”, in terms of square-distance between its vertices. The square-volume function thus described is symmetric in the vertices. The square-volume gives rise to a characterization of the volume form in the top dimension.
متن کاملVolume and Lattice Points of Reflexive Simplices
Using new number-theoretic bounds on the denominators of unit fractions summing up to one, we show that in any dimension d ≥ 4 there is only one d-dimensional reflexive simplex having maximal volume. Moreover, only these reflexive simplices can admit an edge that has the maximal number of lattice points possible for an edge of a reflexive simplex. In general, these simplices are also expected t...
متن کاملMaximum volume simplices and trace-minimal graphs
Let (v, δ) be the set of all δ-regular graphs on v vertices. A graphG ∈ (v, δ) is trace-minimal in (v, δ) if the vector whose ith entry is the trace of the ith power of the adjacency matrix of G, is minimum under the lexicographic order among all such vectors corresponding to graphs in (v, δ). We consider the problem of maximizing the volume of an n-dimensional simplex consisting of n + 1 verti...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 2010
ISSN: 0012-365X
DOI: 10.1016/j.disc.2009.09.026