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Journal of the London Mathematical Society 1993 s2-48(3):385-395; doi:10.1112/jlms/s2-48.3.385
© 1993 by London Mathematical Society
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© Oxford University Press

Two and a Half Remarks on the Marica-Schönheim Inequality

Ron Aharoni and Ron Holzman

Department of Mathematics, Technion-Israel Institute of Technology Haifa 32000, Israel
Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science Rehovot 76100, Israel

The Marica-Schönheim inequality states that the number of distinct differences of the form A\B, with A, B taken from a given finite family A of sets is at least |A|. We prove that equality occurs essentially if and only if A is the product of an ideal and a filter. We also prove an infinite version of the theorem, conjectured (in weaker form) by Daykin and Lovasz. Finally, we note that a generalization (due to Ahlswede and Daykin) of the inequality which considers two families A and B holds under a weaker assumption on the relation between A and B.


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