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


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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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).

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AMS subject classification (1980)

  • 05 C 55
  • 06 A 10