Frederick William Galvin is a mathematician, currently a professor at the

University of Kansas The University of Kansas (KU) is a public research university with its main campus in Lawrence, Kansas, United States, and several satellite campuses, research and educational centers, medical centers, and classes across the state of Kansas. Tw ...

. His research interests include

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

and

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

. His notable combinatorial work includes the proof of the

Dinitz conjecture In combinatorics, the Dinitz theorem (formerly known as Dinitz conjecture) is a statement about the extension of arrays to partial Latin squares, proposed in 1979 by Jeff Dinitz, and proved in 1994 by Fred Galvin. The Dinitz theorem is that given ...

. In set theory, he proved with

András Hajnal András Hajnal (May 13, 1931 – July 30, 2016) was a professor of mathematics at Rutgers University and a member of the Hungarian Academy of Sciences known for his work in set theory and combinatorics. Biography Hajnal was born on 13 May 1931,

that if ℵ_ω₁ is a

strong limit cardinal In mathematics, limit cardinals are certain cardinal numbers. A cardinal number ''λ'' is a weak limit cardinal if ''λ'' is neither a successor cardinal nor zero. This means that one cannot "reach" ''λ'' from another cardinal by repeated success ...

, then :

2^<\aleph_

holds. The research on extending this result led

Saharon Shelah Saharon Shelah ( he, שהרן שלח; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Biography Shelah was born in Jerusalem on July 3, ...

to the invention of

PCF theory PCF theory is the name of a mathematical theory, introduced by Saharon , that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more ap ...

. Galvin gave an elementary proof of the Baumgartner–Hajnal theorem

\omega_1\to(\alpha)^2_k

(

\alpha<\omega_1, k<\omega

). The original proof by Baumgartner and Hajnal used forcing and absoluteness. Galvin and Shelah also proved the square bracket partition relations

\aleph_1\not\to aleph_1 2_4

and

2^\not\to^2_

. Galvin also proved the partition relation

\eta\to

eta Eta (uppercase , lowercase ; grc, ἦτα ''ē̂ta'' or ell, ήτα ''ita'' ) is the seventh letter of the Greek alphabet, representing the close front unrounded vowel . Originally denoting the voiceless glottal fricative in most dialects, ...

2_3 where η denotes the

order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such ...

of the set of rational numbers. Galvin and

Karel Prikry Karel may refer to: People * Karel (given name) * Karel (surname) * Charles Karel Bouley, talk radio personality known on air as Karel * Christiaan Karel Appel, Dutch painter Business * Karel Electronics, a Turkish electronics manufacturer * Gr ...

proved that every

Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...

is Ramsey. Galvin and Komjáth showed that the

axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...

is equivalent to the statement that every graph has a chromatic number. Galvin received his

Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is a ...

in 1967 from the

University of Minnesota The University of Minnesota, formally the University of Minnesota, Twin Cities, (UMN Twin Cities, the U of M, or Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Tw ...

. He invented Doublemove Chess in 1957, and Push Chess in 1967.

References

Living people 20th-century American mathematicians 21st-century American mathematicians Combinatorialists Set theorists University of Minnesota alumni University of Kansas faculty Year of birth missing (living people) {{US-mathematician-stub