We consider the relation between versions of Ramsey’s Theorem and König’s Infinity Lemma, in the absence of the axiom of choice.

In set theory without the Axiom of Choice math formula, we study the deductive strength of the statements math formula, math formula, math formula, and math formula. Among various results, we prove that none of the above statements is provable without using some form of choice, math formula is equivalent to math formula, math formula + math formula implies math formula, math formula does not imply math formula in math formula, math formula does not imply math formula in math formula (...) and math formula is strictly weaker than math formula in math formula. (shrink)