On collection schemes and Gaifman's splitting theorem

Abstract We study model theoretic characterizations of various collection schemes over from the viewpoint of Gaifman's splitting theorem. Among other things, we prove that for any and , the following are equivalent: satisfies the collection scheme for formulas. For any , if , and , then and . For any , if , then . Here, is the unique satisfying . We also investigate strong collection schemes and parameter‐free collection schemes from the similar perspective.