An Explicit Ramsey Bound for de Polignac Numbers

Abstract

A number $m$ is called a \textbf{de Polignac number}($\mathbf{POL}$ in short) if it can be expressed as the difference of infinitely many pairs of consecutive prime numbers. For any given $r\in \mathbb{N},$ a set $A$ is said to be ${\Delta }_r^{\star }$ if for all sets $S$ with $|S|=r$ such that $A\cap \{s-t:s>t\in S\}\ne \varnothing .$ In this article we prove that the set $\mathbf{POL}$ is ${\Delta }_r^{\star }$ with the specific computable value of $r,$ where $r=exp(\mathcal{O}(50)).$ Then we prove that there exists a set $E$ with $|E|\le 2^{50\cdot {\prod }_{i=1}^{49}{p}_i}$ (where $(p_i{)}_i$ is the enumeration of primes) such that $\mathbb{N}={\bigcup }_{t\in E}{t}^{-1}\mathbf{POL}.$