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. Two branch campuses are in the Kansas City metropolitan area on the Kansas side: the university's medical school and hospital ...

. His research interests include

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...

and

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

. His notable combinatorial work includes the proof of the Dinitz conjecture. 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 (; , ; 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, 1945. He is th ...

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 a ...

. 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 Baumgartner (also Baumgärtner, Baumgardner, Bumgardner, Bumgartner or Bumgarner) is a surname of German name, German origin, literally meaning "Tree surgeon, Tree Gardener". It may refer to: ;Baumgartner surname * Ann Baumgartner (1918–2008), fi ...

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 ; ''ē̂ta'' or ''ita'' ) is the seventh letter of the Greek alphabet, representing the close front unrounded vowel, . Originally denoting the voiceless glottal fricative, , in most dialects of Ancient Greek, it ...

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 su ...

of the set of rational numbers. Galvin and

Karel Prikry Karel may refer to: People * Karel (given name) * Karel (surname) * Charles Karel Bouley (born 1962), American talk radio personality known on air as Karel * Christiaan Karel Appel (1921–2006), Dutch painter and sculptor Business * Karel Elec ...

proved that every

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

is Ramsey. Galvin and Komjáth showed that the

axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...

is equivalent to the statement that every

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

has a

chromatic number In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring i ...

. Galvin received his

Ph.D. A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of graduate study and original research. The name of the deg ...

in 1967 from the

University of Minnesota The University of Minnesota Twin Cities (historically known as University of Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Twin Cities of Minneapolis and Saint ...

. He also invented Marseillais chess in 1957 (already published by others earlier in 1925), 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