A set of primes with lower relative density 5/8 and upper density 1 is constructed that fails to represent infinitely many odd integers as sums of three of its elements, showing Shao's 5/8 threshold is sharp.
Alsetri, Density ternary Goldbach for primes in a fixed residue class
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Problems 66 and 67 on sums of residue classes and primes of Andr\'{a}s S\'{a}rk\"{o}zy's collection of unsolved problems
A set of primes with lower relative density 5/8 and upper density 1 is constructed that fails to represent infinitely many odd integers as sums of three of its elements, showing Shao's 5/8 threshold is sharp.