# Fred Galvin

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American mathematician

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**Frederick William Galvin** is a mathematician, currently a professor at the [University of Kansas](/source/University_of_Kansas). His research interests include [set theory](/source/Set_theory) and [combinatorics](/source/Combinatorics).

His notable combinatorial work includes the proof of the [Dinitz conjecture](/source/Dinitz_conjecture). In set theory, he proved with [András Hajnal](/source/Andr%C3%A1s_Hajnal) that if ℵω1 is a [strong limit cardinal](/source/Limit_cardinal), then

- 2 ℵ ω 1 < ℵ ( 2 ℵ 1 ) + {\displaystyle 2^{\aleph _{\omega _{1}}}<\aleph _{(2^{\aleph _{1}})^{+}}}

holds. The research on extending this result led [Saharon Shelah](/source/Saharon_Shelah) to the invention of [PCF theory](/source/PCF_theory). Galvin gave an elementary proof of the [Baumgartner–Hajnal theorem](https://en.wikipedia.org/w/index.php?title=Baumgartner%E2%80%93Hajnal_theorem&action=edit&redlink=1) ω 1 → ( α ) k 2 {\displaystyle \omega _{1}\to (\alpha )_{k}^{2}} ( α < ω 1 , k < ω {\displaystyle \alpha <\omega _{1},k<\omega } ). The original proof by [Baumgartner](/source/James_Earl_Baumgartner) and Hajnal used [forcing](/source/Forcing_(mathematics)) and absoluteness. Galvin and Shelah also proved the square bracket partition relations ℵ 1 ↛ [ ℵ 1 ] 4 2 {\displaystyle \aleph _{1}\not \to [\aleph _{1}]_{4}^{2}} and 2 ℵ 0 ↛ [ 2 ℵ 0 ] ℵ 0 2 {\displaystyle 2^{\aleph _{0}}\not \to [2^{\aleph _{0}}]_{\aleph _{0}}^{2}} . Galvin also proved the partition relation η → [ η ] 3 2 {\displaystyle \eta \to [\eta ]_{3}^{2}} where η denotes the [order type](/source/Order_type) of the set of rational numbers. Galvin and [Karel Prikry](https://en.wikipedia.org/w/index.php?title=Karel_Prikry&action=edit&redlink=1) proved that every [Borel set](/source/Borel_algebra) is Ramsey. Galvin and [Komjáth](/source/P%C3%A9ter_Komj%C3%A1th) showed that the [axiom of choice](/source/Axiom_of_choice) is equivalent to the statement that every [graph](/source/Graph_(discrete_mathematics)) has a [chromatic number](/source/Graph_coloring#Chromatic_number).

Galvin received his [Ph.D.](/source/Ph.D.) in 1967 from the [University of Minnesota](/source/University_of_Minnesota).[1]

He also invented [Marseillais chess](/source/Marseillais_chess) in 1957 (already published by others earlier in 1925), and Push Chess in 1967.

## References

1. **[^](#cite_ref-phd_1-0)** [Fred Galvin](https://mathgenealogy.org/id.php?id=61007) at the [Mathematics Genealogy Project](/source/Mathematics_Genealogy_Project)

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