We introduce the partition function of edge-colored graph homomorphisms, and present an efficient algorithm to approximate it in a certain domain.Expand

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

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

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 be… Expand

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

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 all… Expand

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 partition… Expand

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