Values and bounds for Ramsey numbers associated with polynomial iteration

✍ Scribed by Bruce M. Landman; Raymond N. Greenwell

• Publisher: Elsevier Science
• Year: 1988
• Tongue: English
• Weight: 760 KB
• Volume: 68
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(88)90043-x

✦ Synopsis

Ramsey numbers similar to those of van der Waerden are examined. Rather than considering arithmetic sequences, we look at increasing sequences of positive integers {x1, x2, l l l I x,,} for which there exists a polynomial f(x) = &,aixi, with a, E 2 and Xj+l =f(Xj). We denote by p,(n) the least positive integer such that if [I, 2, . . . , p,(n)] is 2=colored, then there exists a monochromatic sequence of length n generated by a polynomial of degree SC We give values for p,(n) for n s 5, as well as lower bounds for p,(n) and p,(n) We also give an upper bound for certain Ramsey numbers that are in between p,_*(n) and the nth van der Waerden number.