Robert J. Betts

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20102016

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Showing 2012 · math.NTShow all

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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…

math.NT2012

A simple Proof that $r^{k^{2}}$ is an upper Bound on each van der Waerden Number $W (r, k)$ , for each $k$ bounded below by a certain positive real Number $ε$

Robert J. Betts

Here we answer a conjecture by Ron Graham about getting finer upper bounds for van der Waerden numbers in the affirmative, but without the application of double induction or combin…

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 simple Proof that rk2 is an upper Bound on each van der Waerden Number W(r,k), for each k bounded below by a certain positive real Number ε

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 $W (r, k) < r^{k^{2}}$

A simple Proof that $r^{k^{2}}$ is an upper Bound on each van der Waerden Number $W (r, k)$ , for each $k$ bounded below by a certain positive real Number $ε$