Discrete $n$-tuples in Hausdorff spaces

Timothy J. Carlson ¹   ; Neil Hindman ²   ; Dona Strauss ³

¹ Department of Mathematics Ohio State University Columbus, OH 43210, U.S.A.
² Department of Mathematics Howard University Washington, DC 20059, U.S.A.
³ Department of Pure Mathematics University of Hull Hull HU6 7RX, UK

Fundamenta Mathematicae, Tome 187 (2005) no. 2, pp. 111-126

Résumé

We investigate the following three questions: Let $n\in {\mathbb N}$. For which Hausdorff spaces $X$ is it true that whenever ${\mit \Gamma }$ is an arbitrary (respectively finite-to-one, respectively injective) function from ${\mathbb N}^n$ to $X$, there must exist an infinite subset $M$ of ${\mathbb N}$ such that ${\mit \Gamma }[M^n]$ is discrete? Of course, if $n=1$ the answer to all three questions is “all of them”. For $n\geq 2$ the answers to the second and third questions are the same; in the case $n=2$ that answer is “those for which there are only finitely many points which are the limit of injective sequences”. The answers to the remaining instances involve the notion of n-Ramsey limit. We also show that the class of spaces satisfying these discreteness conclusions for all $n$ includes the class of F-spaces. In particular, it includes the Stone–Čech compactification of any discrete space.

Détail

DOI : 10.4064/fm187-2-2

Mots-clés : investigate following three questions mathbb which hausdorff spaces whenever mit gamma arbitrary respectively finite to one respectively injective function mathbb there exist infinite subset mathbb mit gamma discrete course answer three questions geq answers second third questions answer those which there only finitely many points which limit injective sequences answers remaining instances involve notion n ramsey limit class spaces satisfying these discreteness conclusions includes class f spaces particular includes stone ech compactification discrete space