×

Conflict free colorings of nonuniform systems of infinite sets. (English) Zbl 1265.03036

The author gives an alternative proof of Theorem 4.1 of [A. Hajnal et al., Acta Math. Hung. 131, No. 3, 230–274 (2011; Zbl 1274.03074)]: If \(\lambda\) and \(\kappa\) are cardinals and \(\mathcal{H}\) is a subfamily of \([\lambda]^\kappa\) for which there is a fixed natural number \(r\) such that \(| A\cap B| <r\) for all \(A,B\in\mathcal{H}\) then there is a colouring \(c : \lambda\to\omega\) such that for every \(A\in\mathcal{H}\) there is a colour that occurs exactly once in \(A\). In fact the proof works for arbitrary systems of infinite sets, without requiring all members to have the same cardinality.
The author also investigates if it can happen that for every \(A\) all but finitely many colours occur exactly once. He provides a counterexample of cardinality \(\mathfrak{c}\) on a set of cardinality \(\mathfrak{c}\) and shows that if Martin’s Axiom holds for the cardinal \(| \mathcal{H}| \) then the answer is positive, even if the pairwise intersections are only assumed to be finite.

MSC:

03E05 Other combinatorial set theory
03E50 Continuum hypothesis and Martin’s axiom

Citations:

Zbl 1274.03074
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] D. H. Fremlin, Consequences of Martin’s Axiom, Cambridge University Press (1984). · Zbl 0551.03033
[2] A. Hajnal, I. Juhász, L. Soukup and Z. Szentmiklóssy, Conflict free colorings of (strongly) almost disjoint set-systems, Acta Math. Hungar., 131 (2011), 230–274. · Zbl 1274.03074 · doi:10.1007/s10474-010-0051-5
[3] P. J. Szeptycki, Transversals for strongly almost disjoint families, Proc. Amer. Math. Soc., 135 (2007), 2273–2282. · Zbl 1118.03040 · doi:10.1090/S0002-9939-07-08714-X
[4] S. Shelah, Whitehead groups may not be free, even assuming CH, Israel J. Math., 35 (1980), 257–285. · Zbl 0467.03049 · doi:10.1007/BF02760652
[5] L. Soukup, to appear.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.