Polignac's Conjecture in Ramsey-theoretic form

Prove that for every positive integer m, infinitely many consecutive pairs of primes differ by 2m; equivalently, establish the existence of the Ramsey numbers constructed in Theorem \ref{polignac} for all positive integers t and m.

Background

The paper presents Polignac's Conjecture as the assertion that every positive even gap 2m occurs infinitely often between consecutive primes. It constructs a family of generalized Ramsey numbers whose existence for all parameters is equivalent to this conjecture.

The conjecture is explicitly identified as such and is not resolved in the paper. The associated Ramsey-number existence problem is consequently also left unresolved.

References

For every $m\in N$ there are infinitely many values $n\in N$ such that $p_{n+1}-p_n = 2m$.

— Ramsey-Theoretic Characterizations of Classically Non-Ramseyian Problems  (2502.04311 - Christopherson, 6 Feb 2025) in Section 6, Polignac Conjecture and Theorem \ref{polignac}