Constructions and nonexistence results for suitable sets of permutations

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:148

Article

Chan, Justin H. C.¹  Jedwab, Jonathan¹ 

[1] Simon Fraser Univ, Dept Math, 8888 Univ Dr, Burnaby, BC V5A 1S6, Canada

关键词: Construction;    Extremal problem;    Nonexistence;    Ramsey's theorem;    Suitable array;    Suitable core;   

DOI  :  10.1016/j.jcta.2016.12.006

来源: Elsevier

【 摘 要 】

A set of N permutations of {1,2,..., upsilon} is (N, upsilon, t)-suitable if each symbol precedes each subset of t 1 others in at least one permutation. The central problems are to determine the smallest N for which such a set exists for given upsilon and t, and to determine the largest upsilon for which such a set exists for given N and t. These extremal problems were the subject of classical studies by Dushnik in 1950 and Spencer in 1971. We give examples of suitable sets of permutations for new parameter triples (N, upsilon, t). We relate certain suitable sets of permutations with parameter t to others with parameter t+1, thereby showing that one of the two infinite families recently presented by Colbourn can be constructed directly from the other. We prove an exact nonexistence result for suitable sets of permutations using elementary combinatorial arguments. We then establish an asymptotic nonexistence result using Ramsey's theorem. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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