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In 1964, Erd\H{o}s, Hajnal and Moon introduced a saturation version ofTur\'an's classical theorem in extremal graph theory. In particular, theydetermined the minimum number of edges in a $K_r$-free, $n$-vertex graph withthe property that the addition of any further edge yields a copy of $K_r$. Weconsider analogues of this problem in other settings. We prove a saturationversion of the Erd\H{o}s-Szekeres theorem about monotone subsequences andsaturation versions of some Ramsey-type theorems on graphs and Dilworth-typetheorems on posets.  We also consider semisaturation problems, wherein we allow the family to havethe forbidden configuration, but insist that any addition to the family yieldsa new copy of the forbidden configuration. In this setting, we prove asemisaturation version of the Erd\H{o}s-Szekeres theorem on convex $k$-gons, aswell as multiple semisaturation theorems for sequences and posets.

Saturation problems in the Ramsey theory of graphs, posets and point sets