Robert J. Betts

activity

20102016

collaborators

10 papers

cs.DM2016

A Radix Representation for each van der Waerden number $W (r, k)$ with $r$ colors: Why $lo g_{r} W (r, k) < k^{2}$ is true whenever $k$ is the number of terms in the arithmetic progression

Robert J Betts

Here we show that by expressing a van der Waerden number $W (r, k)$ by its radix polynomial representation, it not only is possible to locate each proper subset on $R$ in w…

math.NT2016

Two Upper Bounds for both the van der Waerden Numbers $W (r, k + 1)$ and $W (r + 1, k)$ , that show the Existence of a Recurrence Relation

Robert J Betts

Using a method we have utilized previously, namely through a finite power series expansion which also sometimes is known as the "radix polynomial" representation of an integer, we…

math.NT2016

The Asymptotic Behaviors of $lo g_{r} W (r, k)$ and $lo g_{k} W (r, k)$ , when $W (r, k)$ is a van der Waerden Number

Robert J Betts

We derive the asymptotic behaviors of $lo g_{r} W (r, k)$ and $lo g_{k} W (r, k)$ , when $W (r, k)$ is a van der Waerden Number. We use the approach to consider the subsets on the real l…

cs.DM2016

How to find the least upper bound on the van der Waerden Number $W (r, k)$ that is some integer Power of the coloring Integer $r$

Robert J Betts

What is a least integer upper bound on van der Waerden number $W (r, k)$ among the powers of the integer $r$ ? We show how this can be found by expanding the integer $W (r, k)$ into p…

math.NT2012

A nonconstructive Proof to show the Convergence of the $n^{t h}$ root of diagonal Ramsey Number $r (n, n)$

Robert J. Betts

Does the $n^{t h}$ root of the diagonal Ramsey number converge to a finite limit? The answer is yes. A sequence can be shown to converge if it satifies convergence conditions other…

math.NT2012

The Expansion of each van der Waerden number $W (r, k)$ into Powers of $r$ , when $r$ is the Number of Integer Colorings, determines a greatest lower Bound for all $k$ such that $W (r, k) < r^{k^{2}}$

Robert J. Betts

Every positive integer greater than a positive integer $r$ can be written as an integer that is the sum of powers of $r$ . Here we use this to prove the conjecture posed by Ronald G…

A Radix Representation for each van der Waerden number W(r,k) with r colors: Why logr​W(r,k)<k2 is true whenever k is the number of terms in the arithmetic progression

Two Upper Bounds for both the van der Waerden Numbers W(r,k+1) and W(r+1,k), that show the Existence of a Recurrence Relation

The Asymptotic Behaviors of logr​W(r,k) and logk​W(r,k), when W(r,k) is a van der Waerden Number

How to find the least upper bound on the van der Waerden Number W(r,k) that is some integer Power of the coloring Integer r

A nonconstructive Proof to show the Convergence of the nth root of diagonal Ramsey Number r(n,n)

The Expansion of each van der Waerden number W(r,k) into Powers of r, when r is the Number of Integer Colorings, determines a greatest lower Bound for all k such that W(r,k)<rk2

A Radix Representation for each van der Waerden number $W (r, k)$ with $r$ colors: Why $lo g_{r} W (r, k) < k^{2}$ is true whenever $k$ is the number of terms in the arithmetic progression

Two Upper Bounds for both the van der Waerden Numbers $W (r, k + 1)$ and $W (r + 1, k)$ , that show the Existence of a Recurrence Relation

The Asymptotic Behaviors of $lo g_{r} W (r, k)$ and $lo g_{k} W (r, k)$ , when $W (r, k)$ is a van der Waerden Number

How to find the least upper bound on the van der Waerden Number $W (r, k)$ that is some integer Power of the coloring Integer $r$

A nonconstructive Proof to show the Convergence of the $n^{t h}$ root of diagonal Ramsey Number $r (n, n)$

The Expansion of each van der Waerden number $W (r, k)$ into Powers of $r$ , when $r$ is the Number of Integer Colorings, determines a greatest lower Bound for all $k$ such that $W (r, k) < r^{k^{2}}$