# 無窮元組合學

## 無窮集的拉姆齊理論

${\displaystyle \kappa ,\ \lambda }$序數${\displaystyle m}$基數${\displaystyle n}$為正整數。Erdős & Rado (1956)引入記號

${\displaystyle \kappa \rightarrow (\lambda )_{m}^{n}}$

${\displaystyle m}$${\displaystyle 2}$時，可省略不寫。

${\displaystyle \kappa \rightarrow (\lambda )_{m}^{<\omega }}$

${\displaystyle \aleph _{0}\rightarrow (\aleph _{0})_{k}^{n}}$對所有有限的${\displaystyle n,k}$成立（拉姆齊定理）。
${\displaystyle \beth _{n}^{+}\rightarrow (\aleph _{1})_{\aleph _{0}}^{n+1}}$）。
${\displaystyle 2^{\kappa }\not \rightarrow (\kappa ^{+})^{2}}$謝爾賓斯基定理
${\displaystyle 2^{\kappa }\not \rightarrow (3)_{\kappa }^{2}}$
${\displaystyle \kappa \rightarrow (\kappa ,\aleph _{0})^{2}}$ (）。

${\displaystyle \aleph _{1}\rightarrow (\aleph _{1})_{2}^{\aleph _{1}}.}$

## 大基數

• ${\displaystyle \kappa }$滿足${\displaystyle \kappa \rightarrow (\kappa )^{2}}$
• α艾狄胥基數英语Erdős cardinal${\displaystyle \kappa }$是滿足${\displaystyle \kappa \rightarrow (\alpha )^{\omega }}$的最小基數；
• 拉姆齊基數英语Ramsey cardinal${\displaystyle \kappa }$滿足${\displaystyle \kappa \rightarrow (\kappa )^{\omega }}$

## 參考文獻

1. ^ Blass, Andreas. Ch. 6: Combinatorial Cardinal Characteristics of the Continuum [第6章：連續統的組合基數特徵]. Foreman, Matthew; Kanamori, Akihiro (编). Handbook of Set Theory [集合論手冊]. Springer. 2010 （英语）.
2. ^ Eisworth, Todd. Ch. 15: Successors of Singular Cardinals [第15章：奇異基數的後繼]. Foreman, Matthew; Kanamori, Akihiro (编). Handbook of Set Theory [集合論手冊]. Springer. 2010 （英语）.
• Dushnik, Ben; Miller, E. W., Partially ordered sets [偏序集], American Journal of Mathematics, 1941, 63 (3): 600–610, ISSN 0002-9327, JSTOR 2371374, MR 0004862, doi:10.2307/2371374, （英语）
• Erdős, Paul; Hajnal, András, Unsolved problems in set theory [集合論的未解問題], Axiomatic Set Theory ( Univ. California, Los Angeles, Calif., 1967) [公理化集合論（加州大學，洛杉磯，加州，1967）], Proc. Sympos. Pure Math, XIII Part I, Providence, R.I.: Amer. Math. Soc.: 17–48, 1971, MR 0280381 （英语）
• Erdős, Paul; Hajnal, András; Máté, Attila; Rado, Richard, Combinatorial set theory: partition relations for cardinals [組合集合論：基數的分劃關係], Studies in Logic and the Foundations of Mathematics 106, Amsterdam: North-Holland Publishing Co., 1984, ISBN 0-444-86157-2, MR 0795592 （英语）
• Erdős, P.; Rado, R., A partition calculus in set theory [集合論的分劃算數], Bull. Amer. Math. Soc., 1956, 62 (5): 427–489, MR 0081864, （英语）
• Kanamori, Akihiro. The Higher Infinite: Large Cardinals in Set Theory from their Beginnings [更高的無窮：從源流談集合論的大基數] second. Springer. 2000. ISBN 3-540-00384-3 （英语）.
• Kunen, Kenneth, Set Theory: An Introduction to Independence Proofs [集合論：獨立性證明導論], Amsterdam: North-Holland, 1980, ISBN 978-0-444-85401-8 （英语）