Implications of Ramsey Choice principles in ZF

The Ramsey Choice principle for families of (Formula presented.) -element sets, denoted (Formula presented.), states that every infinite set (Formula presented.) has an infinite subset (Formula presented.) with a choice function on (Formula presented.). We investigate for which positive integers (Formula presented.) and (Formula presented.) the implication (Formula presented.) is provable in (Formula presented.). It will turn out that beside the trivial implications (Formula presented.), under the assumption that every odd integer (Formula presented.) is the sum of three primes (known as ternary Goldbach conjecture), the only non-trivial implication which is provable in (Formula presented.) is (Formula presented.).