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Computing the partition function for graph homomorphisms
TLDR
We introduce the partition function of edge-colored graph homomorphisms, and present an efficient algorithm to approximate it in a certain domain. Expand
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  • 4
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A Generalisation of Tverberg’s Theorem
TLDR
We will prove the following generalisation of Tverberg’s Theorem: given a set S⊂ℝd of (r+1)(k−1)(d+1)+1 points, there is a partition of S in k sets A1, A2,…,Ak such that for any C ⊂S of at most r points, the convex hulls of A1\C,A2\C are intersecting. Expand
  • 24
  • 4
Equal coefficients and tolerance in coloured Tverberg partitions
  • P. Soberón
  • Mathematics, Computer Science
  • Comb.
  • 1 April 2015
TLDR
The coloured Tverberg theorem was conjectured by Bárány, Lovász and Füredi and asks whether for any d+1 sets (considered as colour classes) of k points each in ℝd there is a partition of them into k colourful sets whose convex hulls intersect. Expand
  • 17
  • 3
Helly's Theorem: New Variations and Applications
This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to beExpand
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  • 2
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Computing the partition function for graph homomorphisms with multiplicities
TLDR
We consider a refinement of the partition function of graph homomorphisms and present a quasi-polynomial algorithm to compute it in a certain domain. Expand
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  • 2
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BALANCED CONVEX PARTITIONS OF MEASURES IN ℝ d
We prove the following generalization of the ham sandwich theorem, conjectured by Imre Barany. Given a positive integer k and d nice measures μ 1 , μ 2 ,…, μ d in ℝ d such that μ i (ℝ d )= k for allExpand
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  • 2
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Robust Tverberg and Colourful Carathéodory Results via Random Choice
  • P. Soberón
  • Mathematics, Computer Science
  • Comb. Probab. Comput.
  • 28 June 2016
TLDR
We use the probabilistic method to obtain versions of the colourful Carathéodory theorem and Tverberg's theorem with tolerance. Expand
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  • 2
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Measure partitions using hyperplanes with fixed directions
We study nested partitions of Rd obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partitionExpand
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  • 2
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Equal coefficients and tolerance in coloured tverberg partitions
  • P. Soberón
  • Mathematics, Computer Science
  • SoCG '13
  • 5 April 2012
TLDR
We show that (k-1)d+1 colour classes are necessary and sufficient if the coefficients in the convex combination in the colourful sets are required to be the same in each class. Expand
  • 9
  • 1
Piercing Numbers for Balanced and Unbalanced Families
TLDR
We find an upper bound for the piercing number of families of convex sets with the (p,q)r properties with piercing number one. Expand
  • 12
  • 1
  • PDF