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**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 (\aleph_{c f \alpha} then F in A (\aleph_{\alpha}) (Theorem 7).

If p = \aleph_{\alpha} is singular, r > \aleph_{cf\alpha} q \leq p^+ then [p,q,r] (not)––> s for s \leq p (Theorem 1); for r \leq \aleph_{cf \alpha}, q > \aleph_{cf\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} \aleph_{1}] ––> \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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