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Perfect set property
About: Perfect set property is a research topic. Over the lifetime, 164 publications have been published within this topic receiving 2594 citations.
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TL;DR: In this article, a general theory is developed that leads to a simple criterion for the existence of a perfect code in a distance-transitive graph, and it is shown that this criterion implies Lloyd's theorem in the classical case.
243 citations
01 Jan 1984
TL;DR: In this article, a set of reals X of cardinality ω 1 has universal measure zero if for all measures μ on the Borel sets, there is a Borel set of μ-measure zero covering X.
Abstract: Publisher Summary This chapter discusses some peculiar sets of real numbers and some of the methods for obtaining them. Bernstein constructed a set of reals of cardinality the continuum, which is neither disjoint from nor contains an uncountable closed set. His construction used transfinite induction and the fact that every uncountable closed set has cardinality the continuum. A set of reals is meager if it is the countable union of nowhere dense sets. A set of reals is comeager if it is the complement of a meager set. The Baire category theorem says that no complete metric space is meager in itself. Assuming the continuum hypothesis, there is a set of reals of cardinality the continuum that has countable intersection with every measure zero set. A set of reals X has universal measure zero if for all measures μ on the Borel sets, there is a Borel set of μ-measure zero covering X. The existence of uncountable sets of universal measure zero and uncountable perfectly meager sets does not require any axioms beyond the usual Zermelo–Fraenkel with the axiom of choice. There exists a set of reals X of cardinality ω 1 which has universal measure zero and is perfectly meager.
234 citations
TL;DR: A characterization of perfect zero–one matrices in terms offorbidden submatrices is given, closely related to perfect graphs and constitute a generalization of balanced matrices as introduced by C. Berge.
Abstract: A zero---one matrix is called perfect if the polytope of the associated set packing problem has integral vertices only. By this definition, all totally unimodular zero---one matrices are perfect. In this paper we give a characterization of perfect zero---one matrices in terms offorbidden submatrices. Perfect zero---one matrices are closely related to perfect graphs and constitute a generalization of balanced matrices as introduced by C. Berge. Furthermore, the results obtained here bear on an unsolved problem in graph theory, the strong perfect graph conjecture, also due to C. Berge.
232 citations
Journal Article•
219 citations
TL;DR: It is proved that the category of abelianl-groups is equivalent to the categoryof perfect MV-algebras, and a finite equational axiomatization of the variety generated by perfect MV -algebraes is given.
Abstract: In this paper we prove that the category of abelianl-groups is equivalent to the category of perfect MV-algebras. Furthermore, we give a finite equational axiomatization of the variety generated by perfect MV-algebras.
128 citations