If four subsets of primes satisfy underline delta(P1) + underline delta(P2) > 1 and underline delta(P3) + underline delta(P4) > 1, then every sufficiently large even n is the sum of one prime from each subset, and the bound is sharp.
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A density version of quaternary Goldbach problem
If four subsets of primes satisfy underline delta(P1) + underline delta(P2) > 1 and underline delta(P3) + underline delta(P4) > 1, then every sufficiently large even n is the sum of one prime from each subset, and the bound is sharp.