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Monotone subsequences in (0, 1)-matrices

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Abstract

A (0, 1)-matrix contains anS 0(k) if it has 0-cells (i, j 1), (i + 1,j 2),..., (i + k − 1,j k) for somei andj 1 < ... < jk, and it contains anS 1(k) if it has 1-cells (i 1,j), (i 2,j + 1),...,(i k ,j + k − 1) for somej andi 1 < ... <i k . We prove that ifM is anm × n rectangular (0, 1)-matrix with 1 ≤m ≤ n whose largestk for anS 0(k) isk 0 ≤m, thenM must have anS 1(k) withk ≥ ⌊m/(k 0 + 1)⌋. Similarly, ifM is anm × m lower-triangular matrix whose largestk for anS 0(k) (in the cells on or below the main diagonal) isk 0 ≤m, thenM has anS 1(k) withk ≥ ⌊m/(k 0 + 1)⌋. Moreover, these results are best-possible.

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References

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Authors and Affiliations

  1. Bell Communications Research, 07960, Morristown, NJ, USA

    F. R. K. Chung & V. K. Wei

  2. AT&T Bell Laboratories, 07974, Murray Hill, NJ, USA

    P. C. Fishburn

Authors
  1. F. R. K. Chung
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  2. P. C. Fishburn
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  3. V. K. Wei
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Chung, F.R.K., Fishburn, P.C. & Wei, V.K. Monotone subsequences in (0, 1)-matrices. Graphs and Combinatorics 2, 31–36 (1986). https://doi.org/10.1007/BF01788074

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  • DOI: https://doi.org/10.1007/BF01788074

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