In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán’s classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the Erdős–Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets.

We also consider semisaturation problems, wherein we allow the family to have the forbidden configuration, but insist that any addition to the family yields a new copy of the forbidden configuration. In this setting, we prove a semisaturation version of the Erdős–Szekeres theorem on convex $k$-gons, as well as multiple semisaturation theorems for sequences and posets.

1964年，Erdős，Hajnal和Moon引入了图兰经典定理在极值图论中的饱和版本。特别是，他们确定了${ķ}_{\mathrm{[R}}$-自由， $ñ$-vertex图，具有以下特性：添加任何其他边会产生以下内容的副本： ${ķ}_{\mathrm{[R}}$。我们在其他环境中考虑此问题的类似物。我们证明了关于单调子序列的Erdős-Szekeres定理的饱和版本，以及图上的某些Ramsey型定理和姿势集上的Dilworth型定理的饱和版本。

我们还考虑了半饱和问题，其中我们允许该族具有禁止的配置，但坚持认为，对该族的任何添加都会产生该禁止配置的新副本。在这种情况下，我们证明了凸上的Erdős-Szekeres定理的半饱和版本$ķ$-gons以及序列和位姿的多个半饱和定理。