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Fourier analysis, linear programming, and densities of distance avoiding sets in n

Fernando Mário de Oliveira Filho, Frank Vallentin (2010)

Journal of the European Mathematical Society

We derive new upper bounds for the densities of measurable sets in n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2 , , 24 . This gives new lower bounds for the measurable chromatic number in dimensions 3 , , 24 . We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg,...

Isosceles sets.

Ionin, Yury J. (2009)

The Electronic Journal of Combinatorics [electronic only]

On the number of intersections of two polygons

Jakub Černý, Jan Kára, Daniel Kráľ, Pavel Podbrdský, Miroslava Sotáková, Robert Šámal (2003)

Commentationes Mathematicae Universitatis Carolinae

We study the maximum possible number f ( k , l ) of intersections of the boundaries of a simple k -gon with a simple l -gon in the plane for k , l 3 . To determine the number f ( k , l ) is quite easy and known when k or l is even but still remains open for k and l both odd. We improve (for k l ) the easy upper bound k l - l to k l - k / 6 - l and obtain exact bounds for k = 5 ( f ...

Optimal bounds for the colored Tverberg problem

Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler (2015)

Journal of the European Mathematical Society

We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.

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