= Omega-Jónsson function
{title2=$f:[X]^{\aleph_0}\to X$}
For an infinite <set> $X$, an omega-Jónsson function is a <function>
$$
f:[X]^{\aleph_0}\longrightarrow X
$$
such 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.<Partition calculus (3) by Andrés Caicedo>[Definition 1 and Theorem 2, with the Galvin–Prikry proof]{ref}
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