×

zbMATH — the first resource for mathematics

Unavoidable subprojections in union-closed set systems of infinite breadth. (English) Zbl 07333298
Summary: We consider union-closed set systems with infinite breadth, focusing on three particular configurations \(\mathcal{T}_{\max}(\mathcal{E})\), \(\mathcal{T}_{\min}(\mathcal{E})\) and \(\mathcal{T}_{\operatorname{ort}}(\mathcal{E})\). We show that these three configurations are not isolated examples; in any given union-closed set system of infinite breadth, at least one of these three configurations will occur as a subprojection. This characterizes those union-closed set systems which have infinite breadth, and is the first general structural result for such set systems.
MSC:
20Axx Foundations
03Cxx Model theory
03Gxx Algebraic logic
PDF BibTeX Cite
Full Text: DOI
References:
[1] Aschenbrenner, M.; Dolich, A.; Haskell, D.; Macpherson, D.; Starchenko, S., Vapnik-Chervonenkis density in some theories without the independence property, II, Notre Dame J. Form. Log., 54, 311-363 (2013) · Zbl 1436.03185
[2] Aschenbrenner, M.; Dolich, A.; Haskell, D.; Macpherson, D.; Starchenko, S., Vapnik-Chervonenkis density in some theories without the independence property, I, Trans. Amer. Math. Soc., 368, 5889-5949 (2016) · Zbl 1423.03119
[3] Balogh, J.; Bollobás, B., Unavoidable traces of set systems, Combinatorica, 25, 633-643 (2005) · Zbl 1090.05034
[4] Choi, Y., Approximately multiplicative maps from weighted semilattice algebras, J. Aust. Math. Soc., 95, 36-67 (2013) · Zbl 1317.46033
[5] Y. Choi, M. Ghandehari, H.L. Pham, A construction of non-AMNM weights for every semilattice of infinite breadth, in preparation.
[6] Choi, Y.; Ghandehari, M.; Pham, H. L., Stability of characters and filters for weighted semilattices, Semigroup Forum, 102, 86-103 (2021) · Zbl 07310708
[7] Ditor, S. Z., Cardinality questions concerning semilattices of finite breadth, Discrete Math., 48, 47-59 (1984) · Zbl 0533.06005
[8] Gierz, G., Level sets in finite distributive lattices of breadth 3, Discrete Math., 132, 51-63 (1994) · Zbl 0810.06008
[9] Johnson, B. E., Approximately multiplicative functionals, J. Lond. Math. Soc., 34, 2, 489-510 (1986) · Zbl 0625.46059
[10] Lawson, J. D., The relation of breadth and codimension in topological semilattices. II, Duke Math. J., 38, 555-559 (1971) · Zbl 0243.06004
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.