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Friday, April 15, 2011 |
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Tverberg's theorem and graph coloring |
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Tverberg's theorem is a generalization of Radon's theorem in discrete geometry. It states that any set of N=(d+1)(r-1)+1 points in R^d can be partitioned into r subsets such that the intersection of their convex hulls is non-empty. There are many more ways to partition the points to achieve this than one might imagine at first glance. To illustrate this, one can restrict the allowed subsets of points by introducing a graph on the N points, and force the subsets to be independent. Then the allowed partitions correspond to graph colorings. I will discuss some recent results with Noren on these graph colorings and explain the main conjecture. Most proofs of statements of this type make use of both graph theory and equivariant topology, but I'll mostly talk about the discrete side. |