Abstract
A combinatorial theorem is established, stating that if a familyA 1,A 2, …,A s of subsets of a setM contains every subset of each member, then the complements inM of theA’s have a permutationC 1,C 2, …,C s such thatC i ⊃A i . This is used to determine the minimal size of a maximal set of divisors of a numberN no two of them being coprime.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
De Bruijn, Van Tengbergen, D. Kruiyswijk;On the set of divisors of a number, Nieuw Arch. Wisk.23 (1949–51) 191–193.
J. Schönheim,A generalization of results of P. Erdös, G. Katona, and D. Kleitman concerning Sperner’s theorem, J. Combination Theory (to appear).
G. Katona,A generalization of some generalizations of Sperner’s theorem, J. Combination Theory (to appear).
J. Marica and J. Schönheim,Differences of sets and a problem of Graham, Canad. Math. Bull.12 (1969), 635–638.
P. Erdös, Chao-Ko, R. Rado,Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser.12 (1961), 313–320.
P. Erdös and J. Schönheim;On the set of non pairwise coprime divisors of a number, Proceedings of the Colloquium on Comb. Math. Balaton Füred, 1969 (To appear).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Erdös, P., Herzog, M. & Schönheim, J. An extremal problem on the set of noncoprime divisors of a number. Israel J. Math. 8, 408–412 (1970). https://doi.org/10.1007/BF02798688
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02798688