| Résume | Given a singular cardinal κ and a complete theory T, we say T has exact saturation at κ if there is a κ-saturated model of T that is not κ+-saturated. Stable theories and the random graph have exact saturation at every singular cardinal, while a dense linear order has exact saturation at no singular cardinal–accordingly, failure of exact saturation may be regarded as an avatar of linear order and one might attempt to classify theories according to whether they do or do not have exact saturation. We will describe recent work, joint with Itay Kaplan and Saharon Shelah, which offers several variations on this theme.
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