Splitting strongly almost disjoint families

A. Hajnal, I. Juhasz, S. Shelah

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26 Scopus citations

Abstract

We say that a family A Ì [l]k is strongly almost disjoint if something more than just ïA Ç Bï < k, e.g. that ïA Ç Bï < a < k, is assumed for A, B Î A. We formulate conditions under which every such strongly a.d. family is “essentially disjoint”, i.e. for each A Î A there is F(A) Î [A]<k so that {A \ F(A): A Î A} is disjoint. On the other hand, we get from a supercompact cardinal the consistency of GCH plus the existence of a family A Ì [ww+ 1 ]w 1 whose elements have pairwise finite intersections and such that it does not even have property B. This solves an old problem raised in [4]. The same example is also used to produce a graph of chromatic number w2 on ww+1 that does not contain [w, w], answering a problem from [5]. We also have applications of our results to “splitting” certain families of closed subsets of a topological space. These improve results from [3, 12 and 13].

Original languageEnglish
Pages (from-to)369-387
Number of pages19
JournalTransactions of the American Mathematical Society
Volume295
Issue number1
DOIs
StatePublished - May 1986

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