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Regressions and monotone chains: A ramsey-type extremal problem for partial orders

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Abstract

Aregression is a functiong from a partially ordered set to itself such thatg(x)≦x for allz. Amonotone k-chain is a chain ofk elementsx 1<x 2 <...<x k such thatg(x 1)≦g(x 2)≦...≦g(x k ). If a partial order has sufficiently many elements compared to the size of its largest antichain, every regression on it will have a monotone (k + 1)-chain. Fixingw, letf(w, k) be the smallest number such that every regression on every partial order with size leastf(w, k) but no antichain larger thanw has a monotone (k + 1)-chain. We show thatf(w, k)=(w+1)k.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 61801, Urbana, Illinois, USA

    D. B. West

  2. University of South Carolina, 29208, Columbia, South Carolina, USA

    W. T. Trotter Jr.

  3. Massachusetts Institute of Technology, 02139, Cambridge, Massachusetts, USA

    G. W. Peck & P. Shor

Authors
  1. D. B. West
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  2. W. T. Trotter Jr.
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  3. G. W. Peck
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  4. P. Shor
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Additional information

Dedicated to Paul Erdős on his seventieth birthday

Research supported in part by the National Science Foundation under ISP-80-11451.

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West, D.B., Trotter, W.T., Peck, G.W. et al. Regressions and monotone chains: A ramsey-type extremal problem for partial orders. Combinatorica 4, 117–119 (1984). https://doi.org/10.1007/BF02579164

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  • DOI: https://doi.org/10.1007/BF02579164

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