Authors:
András Hajnal Alfréd Rényi Institute of Mathematics, 1364 Budapest, P.O.B. 127, Hungarye-mails: juhasz@renyi.hu, soukup@renyi.hu

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István Juhász Alfréd Rényi Institute of Mathematics, 1364 Budapest, P.O.B. 127, Hungarye-mails: juhasz@renyi.hu, soukup@renyi.hu

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Lajos Soukup Alfréd Rényi Institute of Mathematics, 1364 Budapest, P.O.B. 127, Hungarye-mails: juhasz@renyi.hu, soukup@renyi.hu

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Zoltán Szentmiklóssy Eötvös Loránd University, 1117 Budapest, Pázmány P. Sétány 1/C., Hungarye-mail: zoli@renyi.hu

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Abstract

is called a conflict free coloring of the set-system(with ρ colors) if
ea
The conflict free chromatic number of is the smallest ρ for which admits a conflict free coloring with ρ colors.
is a (λ,κ,μ)-system if , |A|=κ for all , and is μ-almost disjoint, i.e. |AA′|<μ for distinct . Our aim here is to study
eb
for λκμ, actually restricting ourselves to λω and μω.

For instance, we prove that

• for any limit cardinal κ (or κ=ω) and integers n≧0, k>0, GCH implies
ec

• if λκω>d>1, then λ<κ+ω implies and λ≧ℶω(κ) implies ;

• GCH implies for λκω2 and V=L implies for λκω1;

• the existence of a supercompact cardinal implies the consistency of GCH plus and for 2≦nω;

• CH implies , while implies .

  • [1] Cheilaris, P., Conflict-Free Coloring, PhD thesis, City University of New York (2008).

  • [2] Erdős, P., Galvin, F., Hajnal, A. 1975 On set-systems having large chromatic number and not containing prescribed subsystems Infinite and Finite Sets Colloq. Keszthely 1973 Colloq. Math. Soc. János Bolyai 10 North-Holland Amsterdam 425513 Dedicated to P. Erdős on his 60th birthday, Vol. I.

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  • [3] Erdős, P., Hajnal, A. 1961 On a property of families of sets Acta Math. Acad. Sci. Hung. 12 87124 .

  • [4] Erdős, P., Hajnal, A. 1966 On the chromatic number of graphs and set-systems Acta Math. Acad. Sci. Hung. 17 159229 .

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  • [6] Even, G., Lotker, Z., Ron, D., Smorodinsky, S. 2003 Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellural networks SIAM J. Comput. 33 94136 .

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  • [7] Hajnal, A., Juhász, I., Shelah, S. 1986 Splitting strongly almost disjoint families Trans. Amer. Math. Soc. 295 369387 .

  • [8] Hajnal, A., Juhász, I., Shelah, S. 2000 Strongly almost disjoint families, revisited Fund. Math. 163 1323.

  • [9] Komjáth, P. 1984 Dense systems of almost-disjoint sets Finite and Infinite Sets Eger 1981 Coll Math Soc. J. Bolyai 10 527536.

  • [10] Komjáth, P. 1984 Families close to disjoint ones Acta Math. Hungar. 43 199207 .

  • [11] Kunen, K. 1980 Set Theory North-Holland New York.

  • [12] Miller, E. W. 1937 On a property of families of sets Comptes Rendus Varsovie 30 3138.

  • [13] Pach, J., Tardos, G. 2009 Conflict-free colourings of graphs and hypergraphs Combin. Probab. Comput. 18 819834 .

  • [14] Shelah, S., Diamonds, preprint of paper #922, http://arxiv.org/abs/arXiv:0711.3030.

  • [15] Szeptycki, P. J. 2007 Transversals for strongly almost disjoint families Proc. Amer. Math. Soc. 135 22732282 .

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1950
Volumes
per Year
3
Issues
per Year
6
Founder Magyar Tudományos Akadémia  
Founder's
Address
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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