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ORTHOGONAL COLORINGS OF THE SPHERE
Holmsen, Andreas F.,Lee, Seunghun London Mathematical Society 2016 Mathematika Vol.62 No.2
<P>An orthogonal coloring of the two-dimensional unit sphere $\mathbb{S}^{2}$, is a partition of $\mathbb{S}^{2}$ into parts such that no part contains a pair of orthogonal points: that is, a pair of points at spherical distance ${\it\pi}/2$ apart. It is a well-known result that an orthogonal coloring of $\mathbb{S}^{2}$ requires at least four parts, and orthogonal colorings with exactly four parts can easily be constructed from a regular octahedron centered at the origin. An intriguing question is whether or not every orthogonal 4-coloring of $\mathbb{S}^{2}$ is such an octahedral coloring. In this paper, we address this question and show that if every color class has a non-empty interior, then the coloring is octahedral. Some related results are also given.</P>
Holmsen, Andreas F.,Kincses, Já,nos,Roldá,n-Pensado, Edgardo Elsevier 2016 European journal of combinatorics : Journal europ& Vol.58 No.-
<P><B>Abstract</B></P> <P>We show that for any two convex curves <SUB> C 1 </SUB> and <SUB> C 2 </SUB> in <SUP> R d </SUP> parametrized by [ 0 , 1 ] with opposite orientations, there exists a hyperplane H with the following property: For any t ∈ [ 0 , 1 ] the points <SUB> C 1 </SUB> ( t ) and <SUB> C 2 </SUB> ( t ) are never in the same open half space bounded by H . This will be deduced from a more general result on equipartitions of ordered point sets by hyperplanes.</P>
Near equipartitions of colored point sets
Holmsen, Andreas F.,Kynč,l, Jan,Valculescu, Claudiu Elsevier 2017 Computational geometry Vol.65 No.-
<P><B>Abstract</B></P> <P>Suppose that <I>nk</I> points in general position in the plane are colored red and blue, with at least <I>n</I> points of each color. We show that then there exist <I>n</I> pairwise disjoint convex sets, each of them containing <I>k</I> of the points, and each of them containing points of both colors.</P> <P>We also show that if <I>P</I> is a set of n ( d + 1 ) points in general position in <SUP> R d </SUP> colored by <I>d</I> colors with at least <I>n</I> points of each color, then there exist <I>n</I> pairwise disjoint <I>d</I>-dimensional simplices with vertices in <I>P</I>, each of them containing a point of every color.</P> <P>These results can be viewed as a step towards a common generalization of several previously known geometric partitioning results regarding colored point sets.</P>
The intersection of a matroid and an oriented matroid
Academic Press ; Elsevier Science B.V. Amsterdam 2016 Advances in Mathematics Vol.290 No.-
<P>We prove the following 'matroid intersection' theorem: Let M be a matroid with rank function rho and let O be an oriented matroid of rank r, both defined on the same ground set V and satisfying rho(V) > r. If every subset S subset of V with rho(V \ S) < r contains a positive circuit of O, then there is a positive circuit of O which is independent in M. This contains Imre Barany's colorful Caratheodory theorem as a special case. The proof uses topological methods and combines the Folkman-Lawrence representation theorem with a generalization of Kalai and Meshulam's topological colorful Helly theorem. (C) 2015 Elsevier Inc. All rights reserved.</P>
THE ERDŐS-SZEKERES PROBLEM FOR NON‐CROSSING CONVEX SETS
Dobbins, Michael Gene,Holmsen, Andreas,Hubard, Alfredo London Mathematical Society 2014 Mathematika Vol.60 No.2
<B>Abstract</B><P>We show an equivalence between a conjecture of Bisztriczky and Fejes Tóth about families of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and Tóth on the Erdős-Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk on the Erdős-Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erdős-Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erdős-Szekeres theorem of Pór and Valtr to families of non-crossing convex bodies.</P>