Bruce M. Landman

activity

19992024

most citedSome New Exact van der Waerden Numbers

26 citations · 26 across the 7 of their papers we have counts for

collaborators

7 papers

math.CO2024

Rainbow Numbers for the Generalized Schur Equation $x_{1} + x_{2} + \dots + x_{m - 1} = x_{m}$

Mark Budden, Bruce Landman

We consider the rainbow Schur number $R S_{m} (n)$ , defined to be the minimum number of colors such that every coloring of ${1, 2, \dots, n}$ , using all $R S_{m} (n)$ colors, contains a ra…

math.CO2012

New Bounds on van der Waerden-type Numbers for Generalized 3-term Arithmetic Progressions

Patrick Allen, Bruce M. Landman, Holly Meeks

Let a and b be positive integers with a \leq b. An (a,b)-triple is a set {x,ax+d,bx+ 2d}, where x,d \geq 1. Define T(a,b;r) to be the least positive integer n such that any r-color…

math.CO2007

Bounds on Van der Waerden Numbers and Some Related Functions

Tom Brown, Bruce M. Landman, Aaron Robertson

For positive integers $s$ and $k_{1}, k_{2}, ..., k_{s}$ , let $w (k_{1}, k_{2}, ..., k_{s})$ be the minimum integer $n$ such that any $s$ -coloring ${1, 2, ..., n} \to {1, 2, ..., s}$ admits a $k_{i}$ -…

math.CO2005

On the degree of regularity of generalized van der Waerden triples

Nikos Frantzikinakis, Bruce Landman, Aaron Robertson

Let $1 \leq a \leq b$ be integers. A triple of the form $(x, a x + d, b x + 2 d)$ , where $x, d$ are positive integers is called an {\em (a,b)-triple}. The {\em degree of regularity} of the f…

math.CO2005★ 26 cited

Some New Exact van der Waerden Numbers

Bruce Landman, Aaron Robertson, Clay Culver

For positive integers $r, k_{0}, k_{1}, ..., k_{r - 1},$ the van der Waerden number $w (k_{0}, k_{1}, ..., k_{r - 1})$ is the least positive integer $n$ such that whenever ${1, 2, ..., n}$ is partition…

math.CO2003

Avoiding Monochromatic Sequences With Special Gaps

Bruce M. Landman, Aaron Robertson

For $S$ a set of positive integers, and $k$ and $r$ fixed positive integers, denote by $f (S, k; r)$ the least positive integer $n$ (if it exists) such that within every $r$ -coloring…