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Unions of products of independent sets
Author(s) -
Buczolich Zoltán
Publication year - 1995
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300011311
Subject(s) - mathematics , set (abstract data type) , existential quantification , product (mathematics) , property (philosophy) , combinatorics , discrete mathematics , geometry , computer science , philosophy , epistemology , programming language
We show that there exists an open set H ⊆[0, 1] × [0, 1] with λ 2 ( H ) = 1 such that for any ε > 0 there exists a set E satisfyingλ 1 ( E ) > 1 2 − ε and H contains the product set E × E but there is no set S withλ 1 ( S ) = 1 2and S × S ⊆ H . Especially this property is verified for sets of the form H =∪ i = 1 ∞E i × E iwhere the sets E i are independent andλ 1 ( E i ) < 1 2 . The results of this paper answer questions of M. Laczkovich and are related to a paper of D. H. Fremlin.