In ZFA (Zermelo-Fraenkel set theory with the Axiom of Extensionality weakened to allow the existence of atoms), we prove that the strength of the proposition EDM ("If G = (VG, EG) is a graph such that VG is uncountable, then for every coloring f : [VG] 2 → {0, 1} either there is an uncountable set monochromatic in color 0, or there is a countably infinite set monochromatic in color 1") is strictly between DC ℵ 1 (where DC ℵ 1 is Dependent Choices for ℵ1, a weak choice form stronger than Dependent Choices (DC)) and Kurepa's principle ("Any partially ordered set such that all of its antichains are finite and all of its chains are countable is countable"). Among other new results, we study the relations of EDM to BPI (Boolean Prime Ideal Theorem), RT (Ramsey's theorem), De Bruijn-Erdős' theorem for n-colorings, König's lemma and several other weak choice forms. Moreover, we answer a part of a question raised by Lajos Soukup.

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We examine double successive approximations on a set, which we denote by L 2 L 1 , U 2 U 1 , U 2 L 1 , L 2 U 1 where L 1 , U 1 and L 2 , U 2 are based on generally non-equivalent equivalence relations E 1 and E 2 respectively, on a finite non-empty set V. We consider the case of these operators being given fully defined on its powerset P(V ). Then, we investigate if we can reconstruct the equivalence relations which they may be based on. Directly related to this, is the question of whether there are unique solutions for a given defined operator and the existence of conditions which may characterise this. We find and prove these characterising conditions that equivalence relation pairs should satisfy in order to generate unique such operators.