Jónsson function

with the property that, for any subset y of x with the same cardinality as x, the restriction of

denotes the set of strictly increasing sequences of members of

, using a standard notation for the family of subsets with a given order type.

Erdős and Hajnal (1966) showed that for every ordinal λ there is an ω-Jónsson function for λ. Kunen's proof of Kunen's inconsistency theorem uses a Jónsson function for cardinals λ such that 2λ = λℵ0, and Kunen observed that for this special case there is a simpler proof of the existence of Jónsson functions.

Galvin and Prikry (1976) gave a simple proof for the general case.

The existence of Jónsson functions shows that for any cardinal there is an algebra with an infinitary operation that has no proper subalgebras of the same cardinality.

In particular if infinitary operations are allowed then an analogue of Jónsson algebras exists in any cardinality, so there are no infinitary analogues of Jónsson cardinals.