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introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly ∆ 1−2/k colors. Here ∆ is the maximum degree in the graph and is assumed to be of the order of n δ for some 0 < δ < 1. Their results… (More)

- Ohad Giladi, Assaf Naor, Gideon Schechtman
- 2015

Bourgain's discretization theorem asserts that there exists a universal constant C ∈ (0, ∞) with the following property. Let X, Y be Banach spaces with dim X = n. Fix D ∈ (1, ∞) and set δ = e −n Cn. Assume that N is a δ-net in the unit ball of X and that N admits a bi-Lipschitz embedding into Y with distortion at most D. Then the entire space X admits a… (More)

We show that any L 1 embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid {0, 1,. .. , n} 2 ⊆ R 2 incurs distortion Ω √ log n. We also use Fourier analytic techniques to construct a simple L 1 embedding of this space which has distortion O(log n).

We show that any L 1 embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid {0, 1,. .. , n} 2 ⊆ R 2 incurs distortion Ω log n. We also use Fourier analytic techniques to construct a simple L 1 embedding of this space which has distortion O(log n).

—A major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts. We examine this question in the number-on-the-forehead model of multiparty communication complexity. We show that essentially all… (More)