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A Note on Universal and Canonically Coloured Sequences


A sequence X = {xi}ni=1 over an alphabet containing t symbols is t-universal if every permutation of those symbols is contained as a subsequence. Kleitman and Kwiatkowski showed that the minimum length of a t-universal sequence is (1 − o(1))t2. In this note we address a related Ramsey-type problem. We say that an r-colouring χ of the sequence X is canonical if χ(xi) = χ(xj) whenever xi = xj. We prove that for any fixedt the length of the shortest sequence over an alphabet of size t, which has the property that every r-colouring of its entries contains a t-universal and canonically coloured subsequence, is at most . This is best possible up to a multiplicative constant c independent of r.

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[1]Kleitman, D. J. and Kwiatkowski, D. J. (1976) A lower bound on the length of a sequence containing all permutations as subsequences. J. Combin. Theory Ser. A 21 129136.
[2]Koutas, P. J. and Hu, T. C. (1975) Shortest string containing all permutations. Discrete Math. 11 125132.
[3]Mohanty, S. P. (1980) Shortest string containing all permutations. Discrete Math. 31 9195.
[4]Newey, M. C. (1973) Notes on a problem involving permutations as subsequences. Computer Science Department Report, CS-73-340 Stanford University, Stanford, CA.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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