A GRAPH-THEORETIC APPROACH TO THE UNIQUE MIDSET PROPERTY OF METRIC SPACES

Abstract

A metric space X has the unique midset property if there is a topology-preserving metric d on X such that for every pair of distinct points x, y there is one and only one point p such that d(x, p) = d(y, p). The following are proved. (1) The discrete space with cardinality [nfr ] has the unique midset property if and only if [nfr ] ≠ 2, 4 and [nfr ] [les ] [cfr ], where [cfr ] is the cardinality of the continuum. (2) If D is a discrete space with cardinality not greater than [cfr ], then the countable power D^N of D has the unique midset property. In particular, the Cantor set and the space of irrational numbers have the unique midset property.

A finite discrete space with n points has the unique midset property if and only if there is an edge colouring ϕ of the complete graph K_n such that for every pair of distinct vertices x, y there is one and only one vertex p such that ϕ(xp) = ϕ(yp). Let ump(K_n) be the smallest number of colours necessary for such a colouring of K_n. The following are proved. (3) For each k [ges ] 0, ump(K_2k+1) = k. (4) For each k [ges ] 3, k [les ] ump(K_2k) [les ] 2k−1.