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The following is motivated by an (as of yet) unanswered question on optimal colorings of graphs. I am convinced that the question below has a positive answer in $\newcommand{\ZF}{\text{(ZF)}}\ZF$, but so far it has escaped me.

Let $X\neq\emptyset$ be a set, and let $\newcommand{\Part}{\text{Part}}\Part(X)$ be the collection of partitions of $X$. For $P, Q \in \Part(X)$ we write $P \newcommand{\preq}{\preceq^*}\preq Q$ if there is an injection $\iota: (P \setminus Q) \to (Q \setminus P)$.

Question. If $P,Q,R\in\Part(X)$ with $P\preq Q\preq R$, does this entail that $P\preq R$?

Note. The answer is easily seen to be positive for $P, Q, R$ finite.

The following is motivated by an (as of yet) unanswered question on optimal colorings of graphs. I am convinced that the question below has a positive answer in $\newcommand{\ZF}{{\sf (ZF)}}\ZF$, but so far it has escaped me.

Let $X\neq\emptyset$ be a set, and let $\newcommand{\Part}{\text{Part}}\Part(X)$ be the collection of partitions of $X$. For $P, Q \in \Part(X)$ we write $P \newcommand{\preq}{\preceq^*}\preq Q$ if there is an injection $\iota: (P \setminus Q) \to (Q \setminus P)$.

Question. If $P,Q,R\in\Part(X)$ with $P\preq Q\preq R$, does this entail that $P\preq R$?

Note. The answer is easily seen to be positive for $P, Q, R$ finite.

The following is motivated by an (as of yet) unanswered question on optimal colorings of graphs. I am convinced that the question below has a positive answer in $\newcommand{\ZF}{\text{(ZF)}}\ZF$, but so far it has escaped me.

Let $X\neq\emptyset$ be a set, and let $\newcommand{\Part}{\text{Part}}\Part(X)$ be the collection of partitions of $X$. For $P, Q \in \Part(X)$ we write $P \newcommand{\preq}{\preceq^*}\preq Q$ if there is an injection $\iota: (P \setminus Q) \to (Q \setminus P)$.

Question. If $P,Q,R\in\Part(X)$ with $P\preq Q\preq R$, does this entail that $P\preq R$?

The following is motivated by an (as of yet) unanswered question on optimal colorings of graphs. I am convinced that the question below has a positive answer in $\newcommand{\ZF}{\text{(ZF)}}\ZF$, but so far it has escaped me.

Let $X\neq\emptyset$ be a set, and let $\newcommand{\Part}{\text{Part}}\Part(X)$ be the collection of partitions of $X$. For $P, Q \in \Part(X)$ we write $P \newcommand{\preq}{\preceq^*}\preq Q$ if there is an injection $\iota: (P \setminus Q) \to (Q \setminus P)$.

Question. If $P,Q,R\in\Part(X)$ with $P\preq Q\preq R$, does this entail that $P\preq R$?

Note. The answer is easily seen to be positive for $P, Q, R$ finite.