Erdős, Paul; Hajnal, András On a property of families of sets. (English) Zbl 0201.32801 Acta Math. Acad. Sci. Hung. 12, 87-123 (1961). A family \({\mathcal F}\) of sets is said to have property \(B\) if there exists a set \(S\) meeting all but containing none but one of the members of \({\mathcal F}\), i.e. \(0< |S \cap F| < |F|\) for every \(F \in {\mathcal F}\). The study of property \(B\) was started by E.W.Miller [C. R. Soc. Sci. Varsovie 30, 31-38 (1937; Zbl 0017.30003)]. The authors systematically investigate conditions on a family of sets in order that it does or does not have property \(B\) or a related property. Important examples of the author’s results are the following. (1) Let \({\mathcal F}\) be a family of at most \(\aleph_\omega\) sets, each of power \(\aleph_1\) and with the intersection of any two finite. Then \({\mathcal F}\) has property \(B\). (Whether the result remains true for more than \(\aleph_\omega\) sets is unsettled..) (2) Let \({\mathcal F}\) have \(\aleph_k\) infinite sets, \(k\) finite, any pair of which intersect in at most one element. Then there exists a set \(S\) meeting each in at least one and at most \(k+2\) elements, i.e. \(|S \cap F| \in [1,k+2]\) for all \(F \in {\mathcal F}\), and \(k+2\) is best possible. Many unsolved problems are stated, some of which have since been partially solved in the literature. Soon to be published is “Unsolved problems in set theory” (Axiomatic Set Theory, Proc. Symp. Pure and Applied Math., A.M.S.) in which the same authors summarize the current state of research on these and related problems. Reviewer: M.Krieger Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 17 ReviewsCited in 67 Documents MSC: 05D05 Extremal set theory 03E05 Other combinatorial set theory Citations:Zbl 0017.30003 PDFBibTeX XMLCite \textit{P. Erdős} and \textit{A. Hajnal}, Acta Math. Acad. Sci. Hung. 12, 87--123 (1961; Zbl 0201.32801) Full Text: DOI References: [1] E. W. Miller, On a property of families of sets,Comptes Rendus Varsovie,30 (1937), pp. 31–38. · JFM 63.0832.01 [2] A. Tarski, Sur la décomposition des ensembles en sous ensembles presque disjoint,Fundamenta Math.,14 (1929), pp. 205–215. · JFM 55.0053.04 [3] J. Łos, Linear equations and pure subgroups,Bull. Acad. Polon. Sci. Math.,7 (1959), pp. 13–18. · Zbl 0083.25101 [4] F. Bernstein, Zur Theorie der trigonometrischen Reihen,Leipz. Ber.,60 (1908), pp. 325–338. [5] P. Erdos andA. Hajnal, On the structure of set mappings,Acta Math. Acad. Sci. Hung.,9 (1958), pp. 111–131. · Zbl 0102.28401 [6] A. Hajnal, Some problems and results on set theory,Acta Math. Acad. Sci. Hung.,11 (1960), pp. 277–298. · Zbl 0106.00901 [7] S. Ulam, Zur Maßtheorie in der allgemeinen Mengenlehre,Fundamenta Math.,16 (1930), pp. 140–150. · JFM 56.0920.04 [8] P. Erdos andA. Tarski, On families of mutually exclusive sets,Annals of Math. 44 (1943), pp. 315–329. · Zbl 0060.12602 [9] P. R. Halmos andH. E. Vaughan, The marriage problem,Amer. Journ. Math.,72 (1950), pp. 214–215. · Zbl 0034.29601 [10] N. G. de Bruijn andP. Erdos, A colour problem for infinite graphs and a problem in the theory of relations,Indag. Math.,13 (1951), pp. 371–373. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.