Jónsson function

In mathematical set theory, an ω-Jónsson function for a set x of ordinals is a function ${\displaystyle f:[x]^{\omega }\to x}$ with the property that, for any subset y of x with the same cardinality as x, the restriction of ${\displaystyle f}$ to ${\displaystyle [y]^{\omega }}$ is surjective on ${\displaystyle x}$. Here ${\displaystyle [x]^{\omega }}$ denotes the set of strictly increasing sequences of members of ${\displaystyle x}$, or equivalently the family of subsets of ${\displaystyle x}$ with order type ${\displaystyle \omega }$, using a standard notation for the family of subsets with a given order type. Jónsson functions are named for Bjarni Jónsson.