Twin Prime Conjecture in Ramsey-theoretic form

Establish that infinitely many consecutive pairs of primes differ by 2; equivalently, prove that the Ramsey numbers described in Theorem 5.3 exist for every value of the lower-prime index m.

Background

The paper defines the Twin Prime Conjecture as the assertion that infinitely many consecutive primes have gap 2. It then states that this number-theoretic conjecture is equivalent to the existence, for every m, of a corresponding family of generalized Ramsey numbers whose admissible colorings encode prime distances.

Because the paper explicitly labels the assertion a conjecture and does not prove it, the equivalent Ramsey-theoretic existence question remains unresolved.

References

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

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