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

TL;DR

This paper provides a nonconstructive proof demonstrating that the sequence of the $n^{th}$ roots of diagonal Ramsey numbers converges to a finite limit, specifically to 2, using properties of convergence and complex analysis.

Contribution

It introduces a nonconstructive method to prove the convergence of the $n^{th}$ roots of diagonal Ramsey numbers and characterizes the limit as 2.

Findings

The $n^{th}$ root of diagonal Ramsey numbers converges to a finite limit.

The limit of the $j^{th}$ root of the diagonal Ramsey number is 2.

The convergence can be shown via product representations involving known convergent sequences and complex functions.

Abstract

Does the $n^{th}$ 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 than or besides monotonicity. We show such a property holds for which the sequence of $n^{th}$ roots does converge, even if one has no a priori knowledge as to whether the sequence is monotone or not. We show also the $n^{th}$ root of the diagonal Ramsey number can be expressed as a product of two factors, the first being a known convergent sequence and the second being an absolutely convergent infinite series. One also can express it where one product is convergent and the other has all its values from a uniformly convergent complex function holomorphic within the unit disc on the complex plane. Our motivation solely is to prove the conjecture as a problem in search of a solution, not to establish some deep theory about graphs. A second question is: If the limit exists what is it? At the time of this writing the understanding is the proofs sought need not be constructive. Here we show by nonconstructive proofs that the $n^{th}$ root of the diagonal Ramsey number converges to a finite limit. We also show that the limit of the $j^{th}$ root of the diagonal Ramsey number is two, where positive integer $j$ depends upon the Ramsey number.