Steinhaus Sets and Jackson Sets

2006/03/09 by Su Gao, Gao, Su, Arnold W. Miller +3
Mathematics · #05C12 #11H06 #52C20 #FOS: Mathematics #Logic (math.LO) #Metric Geometry (math.MG) #math.LO #math.MG #msc:05C12 #msc:11H06 #msc:52C20

paper · pdf · doi:10.48550/arxiv.math/0603235

Latex2e: 23 pages Latest version at http://www.math.wisc.edu/~miller

arxiv created 2006/03/09 · arxiv updated 2009/12/01

Abstract

We prove that there does not exist a subset of the plane S that meets every isometric copy of the vertices of the unit square in exactly one point. We give a complete characterization of all three point subsets F of the reals such that there does not exists a set of reals S which meets every isometric copy of F in exactly one point. A finite set X in the plane is Jackson iff for every subset S of the plane there exists an isometric copy Y of X such that Y does not meets S in exactly one point. These results are related to the open problem: Q. (Steve Jackson) Is every finite set X in the plane of two or more points Jackson?

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