Total orders realizable as the distances between two sets of points

2023/04/11 by Maldonado, Gerardo L., Pérez, Miguel Raggi, Roldán-Pensado, Edgardo
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2304.05535

Abstract

In this note we give a negative answer to a question proposed by Almendra-Hernández and Martínez-Sandoval. Let n≤ m be positive integers and let X and Y be sets of sizes n and m in ℝ^n-1 such that every pair of points in X∪ Y defines a unique distance. There is a natural order on X× Y induced by the distances between the corresponding points. The question is if all possible orders on X× Y can be obtained in this way. We show that the answer is negative when n

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