Discrete $n$-tuples in Hausdorff spaces

Volume 187 / 2005

Timothy J. Carlson, Neil Hindman, Dona Strauss Fundamenta Mathematicae 187 (2005), 111-126 MSC: 54A20, 05D10. DOI: 10.4064/fm187-2-2

Abstract

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.

Authors

  • Timothy J. CarlsonDepartment of Mathematics
    Ohio State University
    Columbus, OH 43210, U.S.A.
    e-mail
  • Neil HindmanDepartment of Mathematics
    Howard University
    Washington, DC 20059, U.S.A.
    e-mail
  • Dona StraussDepartment of Pure Mathematics
    University of Hull
    Hull HU6 7RX, UK
    e-mail

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