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Union-Intersecting Set Systems

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Abstract

Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is \(l\)-intersecting. Then we investigate set systems where the union of any \(s\) sets intersect the union of any \(t\) sets. The maximal size of such a set system is determined exactly if \(s+t\le 4\), and asymptotically if \(s+t\ge 5\). Finally, we exactly determine the maximal size of a \(k\)-uniform set system that has the above described \((s,t)\)-union-intersecting property, for large enough \(n\).

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References

  1. Ahlswede, R., Khachatrian, L.H.: The complete intersection theorem for systems of finite sets. Eur. J. Combin. 18, 125–136 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. De Bonis, A., Katona, G.O.H.: Largest families without an r-fork. Order 24, 181–191 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Erdős, P., Ko, C., Rado, R.: Intersection theorems for systems of finite sets. Q. J. Math. Oxford. Second Ser. 12, 313–320 (1961)

    Article  Google Scholar 

  4. Erdős, P., Rado, R.: Intersection theorems for systems of sets. J. Lond. Math. Soc. 35, 85–90 (1960)

    Article  MATH  Google Scholar 

  5. Frankl, P.: On families of finite sets no two of which intersect in a singleton. Bull. Austral. Math. Soc. 17, 125–134 (1977)

  6. Frankl, P.: The Erdős-Ko-Rado theorem is true for n=ckt. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), vol. I, pp. 365–375. Colloq. Math. Soc. János Bolyai, vol. 18. North-Holland, Amsterdam (1978)

  7. Frankl, P., Füredi, Z.: Exact solution of some Turán-type problems. J. Combin. Theory Ser. A 45, 226–262 (1987)

  8. Frankl, P., Füredi, Z.: Beyond the Erdős-Ko-Rado theorem. J. Combin. Theory Ser. A 56(2), 182–194 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Katona, G.: Intersection theorems for systems of finite sets. Acta Math. Acad. Sci. Hungar. 15, 329–337 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  10. Katona, G.O.H., Tarján, T.G.: Extremal problems with excluded subgraphs in the n-cube. Lect. Notes Math. 1018, 84–93 (1981)

    Article  Google Scholar 

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Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15, 1053 , Budapest, Hungary

    Gyula O. H. Katona

  2. Eötvös Loránd University, Egyetem tér 1-3, 1053 , Budapest, Hungary

    Dániel T. Nagy

Authors
  1. Gyula O. H. Katona
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  2. Dániel T. Nagy
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Correspondence to Gyula O. H. Katona.

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Katona, G.O.H., Nagy, D.T. Union-Intersecting Set Systems. Graphs and Combinatorics 31, 1507–1516 (2015). https://doi.org/10.1007/s00373-014-1456-7

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  • DOI: https://doi.org/10.1007/s00373-014-1456-7

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