Bruce Landman

Ramsey Functions for Quasi-Progressions

Abstract. A quasi-progression of diameter n is a finite sequence for which there exists a positive integer L such that for . Let be the least positive integer such that every 2-coloring of will contain a monochromatic k-term quasi-progression of diameter n. We give a lower bound for in terms of k and i that holds for all . Upper bounds are obtained for in all cases for which . In particular, we show that . Exact formulae are found for and . We include a table of computer-generated values of , and make several conjectures.

On the existence of a reasonable upper bound for the van der waerden numbers

Abstract Numbers similar to those of van der Waerden are examined. We consider increasing sequences of positive integers { x 1 , x 2 , …, x n } either that form an arithmetic sequence or for which there exists a polynomial f ( x ) = Σ i = 0 n − 2 a i x i with a i ϵ Z , a n − 2 > 0, and x j + 1 = f ( x j ). We denote by q ( n ) the least positive integer such that if {1, 2, …, q ( n )} is 2-colored, then there exists a monochromatic sequence of the type just described. We give an upper bound for q ( n ), as well as values of q ( n ) for n ⩽ 5. A stronger upper bound for q ( n ) is conjectured and is shown to imply the existence of a similar bound on the n th van der Waerden number.

On generalized van der Waerden triples

Van der Waerdens classical theorem on arithmetic progressions states that for any positive integers k and r, there exists a least positive integer, w(k,r), such that any r-coloring of {1,2 .... ,w(k,r)} must contain a monochromatic k-term arithmetic progression {x,x + d, x + 2d,...,x + (k - 1)d}. We investigate the following generalization of w(3,r). For fixed positive integers a and b with a ≤ b, define N(a,b;r) to be the least positive integer, if it exists, such that any r-coloring of {1,2,...,N(a,b;r)} must contain a monochromatic set of the form {x, ax + d, bx + 2d}. We show that N(a, b; 2) exists if and only if b ≠ 2a, and provide upper and lower bounds for it. We then show that for a large class of pairs (a,b), N(a,b;r) does not exist for r sufficiently large. We also give a result on sets of the form {x,ax + d,ax + 2d ..... ax + (k - 1 )d}.

Generalized van der Waerden numbers

Certain generalizations of arithmetic progressions are used to define numbers analogous to the van der Waerden numbers. Several exact values of the new numbers are given, and upper bounds for these numbers are obtained. In addition, a comparison is made between the number of different arithmetic progressions and the number of different generalized arithmetic progressions.

Values and bounds for Ramsey numbers associated with polynomial iteration

Abstract 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,…,xn} for which there exists a polynomial ⨍(x)=∑ r i =0 a i x i , with aiϵZ and x j +1 =⨍(x j ) . We denote by pr(n) the least positive integer such that if [1,2,…,pr(n)] is 2-colored, then there exists a monochromatic sequence of length n generated by a polynomial of degree ⩽r. We give values for pr(n) for n⩽5, as well as lower bounds for p1(n) and p2(n). We also give an upper bound for certain Ramsey numbers that are in between pn−2(n) and the nth van der Waerden number.

Avoiding Monochromatic Sequences With Special Gaps

For

ON SOME GENERALIZATIONS OF THE VAN DER WAERDEN NUMBER W(3)

S \subseteq \mathbb{Z}^+

Monochromatic Arithmetic Progressions With Large Differences

and

An upper bound for van der Waerden-like numbers usingk colors

k

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

and