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A Graph‐Theoretic Approach to the Unique Midset Property of Metric Spaces
Author(s) -
Itō Munehiko,
Ohta Haruto,
Ono Jin
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799007887
Subject(s) - mathematics , combinatorics , cardinality (data modeling) , vertex (graph theory) , metric space , discrete space , discrete mathematics , countable set , complete metric space , space (punctuation) , graph , irrational number , property (philosophy) , computer science , mathematical analysis , geometry , epistemology , data mining , operating system , philosophy
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 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 ⩾ 0, ump( K 2 k +1 ) = k . (4) For each k ⩾ 3, k ⩽ ump( K 2 k ) ⩽ 2 k −1.