Graphs and Combinatorics

, Volume 6, Issue 3, pp 287–291 | Cite as

Some new bounds and values for van der Waerden-like numbers

  • Bruce M. Landman
  • Raymond N. Greenwell


Numbers similar to those of van der Waerden are studied. We consider increasing sequences of positive integers {x1,x2,...,x n } that either form an arithmetic progression or for which there exists a polynomialf with integer coefficients and degree exactlyn − 2, andxj+1 =f(x j ). We denote byq(n, k) the least positive integer such that if {1, 2,...,q(n, k)} is partitioned intok classes, then some class must contain a sequence of the type just described. Upper bounds are obtained forq(n, 3), q(n, 4), q(3, k), andq(4, k). A table of several values is also given.


Positive Integer Arithmetic Progression Integer Coefficient 
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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Bruce M. Landman
    • 1
  • Raymond N. Greenwell
    • 2
  1. 1.Department of MathematicsUniversity of North Carolina at GreensboroGreensboroUSA
  2. 2.Department of MathematicsHofstra UniversityHempsteadUSA

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