Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 102.28501
Autor: Erdös, Pál; Hajnal, András
Title: Some remarks on set theory. VII. (In English)
Source: Acta Sci. Math. 21, 154-163 (1960).
Review: [Part VI: Zbl 078.04203]
For a family F of sets let pF = \sup |X| (X in F). For a cardinal r the authors say that F has the property A(r) if some set X of cardinality < r intersects every member of F: one could write F in A(r). Analogously, F in A(q,r) if F' \subset F, |F'| < q ==> F' in A(r). Def.: (1) [p,q,r] ––> s means that every F such that pF = p, F in A(q,r) possesses the property A(s) too. F possesses the property B(t) if every F' \subseteq F with |F'| = t has a subfamily F'' of cardinality t such that \bigcap F'' is non empty. Let p = p(F); F in B(p) ==> F in A(p)(Theorem 6); if moreover p = \aleph\alpha is singular and F in B (\alephc f \alpha then F in A (\aleph\alpha) (Theorem 7).
If p = \aleph\alpha is singular, r > \alephcf\alpha q \leq p^+ then [p,q,r] (not)––> s for s \leq p (Theorem 1); for r \leq \alephcf \alpha, q > \alephcf\alpha one has [p,q,r] ––> p (Theorem 2). If \aleph\alpha is regular, then [\aleph\alpha+n, \aleph\alpha+n+1, \aleph\alpha ] ––> \aleph\alpha (Theorem 5).
Problem 1. Does \aleph\omega+1\aleph0 \leq 2\aleph0. \aleph\omega+1 imply [\aleph\omega+1 \aleph\omega+2 \aleph1] ––> \aleph\omega? There are also other results, concerning the relation (1), in particular implied by the general continuum hypothesis.
Reviewer: G.Kurepa
Classif.: * 05D10 Ramsey theory
Index Words: set theory
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