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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Splitting strongly almost disjoint families


Authors: A. Hajnal, I. Juhász and S. Shelah
Journal: Trans. Amer. Math. Soc. 295 (1986), 369-387
MSC: Primary 03E05; Secondary 03E35, 03E55, 04A20, 54A25, 54A35
DOI: https://doi.org/10.1090/S0002-9947-1986-0831204-9
MathSciNet review: 831204
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Abstract: We say that a family $ \mathcal{A} \subset {[\lambda ]^\kappa }$ is strongly almost disjoint if something more than just $ \vert A \cap B\vert < \kappa $, e.g. that $ \vert A \cap B\vert < \sigma < \kappa $, is assumed for $ A$, $ B \in \mathcal{A}$. We formulate conditions under which every such strongly a.d. family is "essentially disjoint", i.e. for each $ A \in \mathcal{A}$ there is $ F(A) \in {[A]^{ < \kappa }}$ so that $ \{ A\backslash F(A):A \in \mathcal{A}\} $ is disjoint. On the other hand, we get from a supercompact cardinal the consistency of $ {\text{GCH}}$ plus the existence of a family $ \mathcal{A} \subset {[{\omega _{\omega + 1}}]^{{\omega _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 $ {\omega _2}$ on $ {\omega _{\omega + 1}}$ that does not contain $ [\omega ,\omega ]$, 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 [ $ {\mathbf{3}},{\mathbf{12}}$ and $ {\mathbf{13}}$].


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DOI: https://doi.org/10.1090/S0002-9947-1986-0831204-9
Article copyright: © Copyright 1986 American Mathematical Society