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Avoiding Arrays of Odd Order by Latin Squares

Published online by Cambridge University Press:  21 December 2012

LINA J. ANDRÉN ,
CARL JOHAN CASSELGREN  and
LARS-DANIEL ÖHMAN
LINA J. ANDRÉN
Affiliation:
Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187 Umeå, Sweden (e-mail: andren.lina@gmail.com, lars-daniel.ohman@math.umu.se)
CARL JOHAN CASSELGREN
Affiliation:
Department of Mathematics, Linköping University, SE-58183 Linköping, Sweden (e-mail: carl.johan.casselgren@liu.se)
LARS-DANIEL ÖHMAN
Affiliation:
Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187 Umeå, Sweden (e-mail: andren.lina@gmail.com, lars-daniel.ohman@math.umu.se)
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Abstract

We prove that there is a constant c such that, for each positive integer k, every (2k + 1) × (2k + 1) array A on the symbols (1,. . .,2k+1) with at most c(2k+1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1) × (2k+1) Latin square S on the symbols 1,. . .,2k+1 such that, for each i,j ∈ {1,. . .,2k+1}, the symbol in position (i,j) of S does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist. Using this result, we also show that there is a constant ρ, such that, for any positive integer n, if each cell in an n × n array B is assigned a set of m ≤ ρ n symbols, where each set is chosen independently and uniformly at random from {1,. . .,n}, then the probability that B is avoidable tends to 1 as n → ∞.

Keywords

Primary 05B15Secondary 05C15

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 2 , March 2013 , pp. 184 - 212
Copyright
Copyright © Cambridge University Press 2012

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