Omega-Jónsson function ($f:[X{]}^{{\aleph }_0}\to X$)

For an infinite set $X$, an omega-Jónsson function is a functionsuch that $f([Y{]}^{{\aleph }_0})=X$ for every subset $Y\subseteq X$ with $|Y|=|X|$. Here $[X{]}^{{\aleph }_0}$ denotes all countably infinite subsets of $X$. Thus deleting any collection of elements while preserving the cardinal number of $X$ still leaves inputs attaining every original output value.

For example, when $X=\mathbb{N}$, the rule $f(A)=\min A$ fails: restricting inputs to infinite subsets of the even numbers loses every odd output. An omega-Jónsson function must distribute its values much less locally. Under the axiom of choice, such a function exists for every infinite set, a theorem of Paul Erdős and András Hajnal. The countably infinite input size is essential to this assertion; it is a strong negative partition relation rather than an ordinary finite coloring statement.^[ref]