# Optimal partitions amongst a given set of partitions

Let $$X\neq \emptyset$$ be a set. By $$\text{Part}(X)$$ we denote the set of all partitions of $$X$$ not containing $$\emptyset$$ as an element.

First, note that $$\bigcup{\frak P}$$ is the collection of subsets of $$X$$ that are a member of at least one partition in $${\frak P}$$. We say that $${\frak P}\subseteq \text{Part}(X)$$ is admissible if it is closed under refinement and if for all $$b\in \bigcup {\frak P}$$ there is $$b_0\in \bigcup{\frak P}$$ such that $$b \subseteq b_0$$, and $$b_0$$ is maximal in $$\bigcup{\frak P}$$ with respect to set inclusion. Moreover, we say that $${\frak P}$$ has the optimality property if there is $$P_0\in {\frak P}$$ such that $$|P_0\setminus P| \leq |P \setminus P_0|$$ for all $$P\in {\frak P}$$.

Given $${\frak P}\subseteq \text{Part}(X)$$ as well as a non-empty set $$S\subseteq X$$, we let $${\frak P}|_S$$ be $${\frak P}$$ "cut down to $$S$$", or more formally, $${\frak P}|_S := \{P \in \text{Part}(S): (\exists Q \in \text{Part}(X))(\forall b\in P)(\exists b' \in Q) b = b'\cap S\}.$$

Let $$X$$ be non-empty and $${\frak P}\subseteq \text{Part}(X)$$ be admissible. Suppose that $${\cal C}$$ is a collection of subsets of $$X$$ such that for $$C, C'\in {\cal C}$$ we have $$C\subseteq C'$$ or $$C'\subseteq C$$, and also we have $$\bigcup {\cal C} = X$$. Suppose further that for every set $$C\in {\cal C}$$, the collection $${\frak P}|_C \subseteq \text{Part}(C)$$ has the optimality property.

Question. Does this imply that $$\frak P$$ itself has the optimality property?

Note. Without the maximality requirement for admissibility, there would be an easy negative answer for the question. Let $$X = \omega$$, and let $${\frak P}$$ consist of the partitions of $$\omega$$ such that every block (= member of the partition) is finite. Let $${\cal C}$$ consist of the sets of the form $$n := \{0, \ldots, n-1\}$$ for all positive integers $$n$$. Then the union of $${\cal C}$$ equals $$\omega$$, and $${\frak P}|_n$$ trivially satisfies the optimality property, but $${\frak P}$$ itself does not.

Let $$X=\{0,1,2,\ldots\}$$ and let $$P=\{\{0,1\},\{2,3\},\{4,5\},\ldots\}$$. Let $$\mathfrak{P}$$ be all partitions of $$X$$ which refine $$P$$ and moreover contain only finitely many of the blocks in $$P$$ (thus for instance $$P$$ itself does not belong). Cover $$X$$ by sets of the form $$\{0,1,\ldots,n\}$$, like in your note. Now I think $$\mathfrak{P}$$ is admissible but still fails the optimality property.