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András Hajnal was a mathematician working in set theory and combinatorics. He served as director of DIMACS and later as a Rutgers mathematics professor before returning to Hungary after retirement.^[ref]

One result bearing his name, proved with Paul Erdős, says that every infinite set admits an omega-Jónsson function: a colouring of its countably infinite subsets by its elements such that every subset of the same cardinality still sees every colour. Assuming the axiom of choice, even restricting to a large subset cannot simplify this colouring by omitting a colour. This illustrates the negative side of infinite partition relations, complementing Ramsey theory results that force homogeneous subsets.^[ref]

Andrés Caicedo is the mathematician specializing in set theory identified as Andrés Eduardo Caicedo on his author homepage.^[ref] His published teaching notes develop infinite partition relations, including omega-Jónsson functions and their relationship to colourings of countably infinite subsets.^[ref]

For an infinite set $X$, an omega-Jónsson function has the form $f:[X{]}^{{\aleph }_0}\to X$, and every $Y\subseteq X$ with $|Y|=|X|$ satisfies $f([Y{]}^{{\aleph }_0})=X$. Thus a large restriction still produces every colour. His notes present a proof of existence assuming the axiom of choice, explicitly attributing the theorem and the simplified proof to their discoverers. This is a concrete example of the interaction between cardinal arithmetic and infinite combinatorics.

Let $j:V\to M$ be a nonidentity elementary embedding into a transitive class model of ZFC, with its set restrictions and Kunen critical sequence available. Let $\lambda ={\sup }_{n<\omega }{j}^n(\kappa )$ for $\kappa =\mathrm{crit}(j)$. Then $j(\lambda )=\lambda$ andThese domain and closure assumptions are essential; the assertion is not a statement about an arbitrary elementary map between arbitrary structures.^[ref]

For the combinatorial argument, choose an omega-Jónsson function $f:[\lambda {]}^{\omega }\to \lambda$: every subset of $\lambda$ of cardinality $\lambda$ has all $\lambda$ possible values on its countable subsets. If $X=j‘‘\lambda$ belonged to $M$, its order type would be $\lambda$, so $M$ would see it as a set of cardinality $\lambda$. Elementarity would make $j(f)$ take every value below $\lambda$ on its internally countable subsets. But every such subset $Y\subseteq X$ is $j(Z)$ for a countable $Z\subseteq \lambda$, since $j$ fixes countable enumerations coordinatewise. HenceThe value $\kappa$ is omitted by that image, a contradiction. The existence of omega-Jónsson functions is the combinatorial input; the rest tracks precisely what the target model could contain.