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This is a survey paper on rainbow sets (another name for "choice functions").The main theme is the distinction between two types of choice functions: those having a large (in the sense of belonging to some specified filter, namely closed-up set) image, and those that have a large domain and small image, where "smallness" means belonging to some specified complex (a closed-down set).The paper contains some new results: (1) theorems on scrambled versions, in which the sets are re-shuffled before choosing the rainbow set, and (2) results on weighted and cooperative versions -defined in the article.
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Aharoni et al. (2024) studied this question.
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