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Journal of the London Mathematical Society 1999 60(2):353-365; doi:10.1112/S0024610799007887
© 1999 by London Mathematical Society
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© The London Mathematical Society

A Graph-Theoretic Approach to the Unique Midset Property of Metric Spaces

Munehiko Ito, Haruto Ohta and Jin Ono

Faculty of Science, Kochi University Kochi 780, Japan
Faculty of Education, Shizuoka University Ohya, Shizuoka 422, Japan
Faculty of Engineering, Shizuoka University Ohya, Shizuoka 422, Japan

Received 9 July 1996.

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 n has the unique midset property if and only if n != 2, 4 and n ≤ c, where c is the cardinality of the continuum. (2) If D is a discrete space with cardinality not greater than c, then the countable power DN 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 {varphi} of the complete graph Kn such that for every pair of distinct vertices x, y there is one and only one vertex p such that {varphi}(xp) = {varphi}(yp). Let ump(Kn) be the smallest number of colours necessary for such a colouring of Kn. The following are proved. (3) For each k ≥ 0, ump(K2k+1) = k. (4) For each k ≥ 3, k ≤ ump(K2k) ≤ 2k–1.


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