Partial order on a family of κ-subsets of a linearly ordered set
For k-subsets A,B of the rationals Q, define A≻nB if a>b holds for at least n ordered pairs (a,b)∈A×B, where k,n are integers, 1≤n≤k2. We prove that (1) the relation ≻n is transitive if and only if k2-k+1≤n, and (2) there is a cyclic sequence A1≻nA2≻n⋯≻ nAr≻nA1 of k-subsets of Q if and only if 1≤...
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| Format: | text |
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Animo Repository
2006
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| Online Access: | https://animorepository.dlsu.edu.ph/faculty_research/1372 https://animorepository.dlsu.edu.ph/context/faculty_research/article/2371/type/native/viewcontent |
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| Institution: | De La Salle University |
| Summary: | For k-subsets A,B of the rationals Q, define A≻nB if a>b holds for at least n ordered pairs (a,b)∈A×B, where k,n are integers, 1≤n≤k2. We prove that (1) the relation ≻n is transitive if and only if k2-k+1≤n, and (2) there is a cyclic sequence A1≻nA2≻n⋯≻ nAr≻nA1 of k-subsets of Q if and only if 1≤n≤k2-⌊(k+1)2/ 4⌋. We also investigate the length of such cyclic sequences. © 2006 Elsevier B.V. All rights reserved. |
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