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Separability properties of almost-disjoint families of sets. (English) Zbl 0246.05002

Several results are proved on almost disjoint sets and many new problems are raised. Among others the authors prove the following conjecture of Hechler: Let \(F\) be a family of infinite sets \(\{A_\alpha \}\) satisfying \(|A_{\alpha_1} \cap A_{\alpha_2}| < \aleph_0\). Assume further that the family has chromatic number \(>2\) (or does not have property \(B\)) i.e. if a set has a non-empty intersection with every \(A\), then it contains at least one of them. Then there must be two \(A\)’s which are disjoint. Further in general it is not true that there are three \(A\)’s which are pairwise disjoint.

MSC:

05A05 Permutations, words, matrices
03E05 Other combinatorial set theory
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References:

[1] P. Erdös andA. Hajnal,On a property of families of sets, Acta Math. Acad. Sci. Hungar.12 (1961), 87–123. · Zbl 0201.32801
[2] P. Erdös andA. Hajnal,On chromatic number of graphs and set systems, Acta Math. Acad. Sci. Hungar.17 (1966), 61. · Zbl 0151.33701
[3] S. H. Hechler,Classifying a almost-disjoint families with applications to {\(\beta\)}N-N, Israel J. Math.10 (1971), 413–432. · Zbl 0232.04003
[4] D. M. Martin andM. Solovay,Internal Cohen extensions, Annals of Math. Logic2 (1970), 143–178. · Zbl 0222.02075
[5] E. W. Miller,On a property of families of sets, Comptes Rendus Varsovie,30 (1937) 31–38. · Zbl 0017.30003
[6] S. H. Hechler,Short complete nested sequences in {\(\beta\)}N-N and small almost-disjoint families, to appear in General Topology and its Appl. · Zbl 0246.02047
[7] P. Erdös,On a combinatorial problems II, Acta Math. Acad. Sci. Hungar.15 (1964), 445–447. · Zbl 0201.33704
[8] W. M. Schmidt,Ein Kombinatorisches Problem von P. Erdös und A. Hajnal., Acta Math. Acad. Sci. Hungar.15 (1964), 373–374. · Zbl 0143.02501
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