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 is strongly almost disjoint if something more than just , e.g. that , is assumed for , . We formulate conditions under which every such strongly a.d. family is "essentially disjoint", i.e. for each there is so that is disjoint. On the other hand, we get from a supercompact cardinal the consistency of plus the existence of a family whose elements have pairwise finite intersections and such that it does not even have property . This solves an old problem raised in [**4**]. The same example is also used to produce a graph of chromatic number on that does not contain , 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 [ and ].

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DOI:
https://doi.org/10.1090/S0002-9947-1986-0831204-9

Article copyright:
© Copyright 1986
American Mathematical Society