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

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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References

  1. [1]

    E. Harzheim, Combinatorial theorems on contractive mappings in power sets,Discrete Math. 40 (1982), 193–201.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    R. Rado, A theorem on chains of finite sets, II,Acta Arithmetica 43 (1971), 257–261.

    MathSciNet  Google Scholar 

  3. [3]

    F. P. Ramsey, On a problem of formal logic,Proc. London Math. Soc. (2),30 (1930), 264–286.

    Article  Google Scholar 

  4. [4]

    D. B. West, Extremal problems in partially ordered sets, in:Ordered Sets, (Ivan Rival, ed.), D. Reidel Publishing Co. (1982), 473–521.

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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). https://doi.org/10.1007/BF02579164

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AMS subject classification (1980)

  • 05 C 55
  • 06 A 10