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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]

Nicolaas Govert de Bruijn worked on combinatorics and mathematical structures. With Paul Erdős, he proved a finite incidence theorem in their 1948 paper: a nontrivial system in which each pair of points lies on exactly one line has at least as many lines as points.^[ref]

The equality cases are highly organized: a near-pencil or a finite projective plane. This shows how a local pair-incidence rule controls both a global counting inequality and the structure of extremal examples.^[ref]

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]

By Nicolaas Govert de Bruijn and Paul Erdős. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen51,1277–1279 (also Indagationes Mathematicae10,421–423). Primary scan: users.renyi.hu/~p_erdos/1948-01.pdf

Theorem1 gives the finite unique-pair line-count bound and identifies the near-pencil and projective-plane equality configurations.

Paul Erdős worked on number theory, combinatorics and extremal problems. His joint 1948 paper with Nicolaas Govert de Bruijn gives an incidence bound from a unique-line-through-two-points condition.^[ref]

For a finite nontrivial incidence system, the number of lines is at least the number of points. Equality forces a near-pencil or a finite projective plane. The theorem illustrates an extremal approach: determine a sharp numerical bound, then understand the configurations attaining it.^[ref]