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By constructing special relation algebras we show that if 3<α<ω, then SNr 3 CA α ¬=;SNr 3 CA 3α−7 and there is a logically valid first-order sentence containing at most three variables with a proof in which every sentence has at most 3α−7 variables, but no proof in which every sentence has at most α variables

The neat embedding problem and the number of variables required in proofs