For a set $X$, let $|X|$ denote its cardinality.
A block of a partition is a non-empty element of the partition.



> 
> Let $P$ and $Q$ be two partitions of a set $X$.
>  If $|P| < |Q|$ then $P$ contains a block $B$ which intersects two distinct blocks of $Q$.
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What is this principle called?


The proof is immediate for finite $X$, and is an easy consequence of the Axiom of Choice for infinite $X$. I am interested in using this result, and would like to refer to its proper name and to cite it correctly. Jech's *Set Theory* and several online sources don't seem to mention this result. In particular the [Consequences of the Axiom of Choice project](http://consequences.emich.edu/conseq.htm) doesn't seem to list it (at least not in this specific form).


I would also be interested to know whether this principle is equivalent to some known consequence of the Axiom of Choice.



Name for this generalized pigeonhole principle?
This is equivalent to the Weak Partition Principle (a close relative of the Partition Principle mentioned by Goldstern). The Weak Partition Principle is form 100 in *Consequences of the Axiom of Choice* by Howard and Rubin (the Partition Principle is form 101).


The Weak Partition Principle asserts that if there is a surjection from $X$ onto $Y$ then it is not the case that $X$ has strictly smaller cardinality than $Y$ (i.e. $X \prec Y$). In contrapositive form, if $X \prec Y$ then there is no surjection from $X$ onto $Y$.


To see that the Weak Partition Principle implies your statement, note that if $P$ contains no block which intersects two distinct blocks of $Q$ then each block of $P$ is contained in a unique block of $Q$ and the map $P \to Q$ thus defined is necessarily a surjection. By the Weak Partition Principle, this cannot hold at the same time as the hypothesis $P \prec Q$.


For the converse, suppose that the Weak Partition Principle fails as witnessed by sets $X \prec Y$ and a surjection $p:X \to Y$. Let $Q = \lbrace p^{-1}(y) : y \in Y \rbrace$ and $P = \lbrace \lbrace x \rbrace : x \in X\rbrace$. These are two partitions of $X$ with $P \prec Q$ and every block of $P$ is clearly contained in a unique block of $Q$.


