Abstract
Let \({\mathcal{F}}\) be a family of subsets of an n-element set. \({\mathcal{F}}\) is called (p,q)-chain intersecting if it does not contain chains \(A_1\subsetneq A_2\subsetneq\dots\subsetneq A_p\) and \(B_1\subsetneq B_2\subsetneq\dots\subsetneq B_q\) with \(A_p\cap B_q=\emptyset\) . The maximum size of these families is determined in this paper. Similarly to the p = q = 1 special case (intersecting families) this depends on the notion of r-complementing-chain-pair-free families, where r = p + q − 1. A family \({\mathcal{F}}\) is called r-complementing-chain-pair-free if there is no chain \({\mathcal{L}} \subseteq {\mathcal{F}}\) of length r such that the complement of every set in \({\mathcal{L}}\) also belongs to \({\mathcal{F}}\) . The maximum size of such families is also determined here and optimal constructions are characterized.
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References
Bernáth, A.: On a problem in extremal set theory, Mat. Lapok (N.S.) 10 (2000/01) (2), 2–4 (2005)
de Bruijn, N.G., van Ebbenhorst Tengbergen, Ca., Kruyswijk, D.: On the set of divisors of a number, Nieuw Arch. Wiskunde 23(2), 191–193 (1951)
Erdős, P.: On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc. 51, 898–902 (1945)
Erdős, P., Ko, C., Rado, R.: Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. 12(2), 313–320 (1961)
Erdős, P.L., Füredi, Z., Katona, G.O.H.: Two-part and k-Sperner families: new proofs using permutations, SIAM J. Discrete Math. 19(2), 489–500 (electronic) (2005)
Gerbner, D.: On an extremal problem of set systems, Mat. Lapok (N.S.) 10 (2000/01), (2), 5–12 (2005)
Katona, G.O.H.: A simple proof of the Erdős-Chao Ko-Rado theorem, J. Comb. Theory Ser. B 13, 183–184 (1972)
Lubell, D.: A short proof of Sperner’s lemma, J. Comb. Theory 1, 299 (1966)
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The first author is a member of the Egerváry Research Group (EGRES). Research is supported by OTKA grants T 037547 and TS 049788, by European MCRTN Adonet, Contract Grant No. 504438 and by the Egerváry Research Group of the Hungarian Academy of Sciences.
The work of the second author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA), grant numbers T037846 and NK62321.
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Bernáth, A., Gerbner, D. Chain Intersecting Families. Graphs and Combinatorics 23, 353–366 (2007). https://doi.org/10.1007/s00373-007-0743-y
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DOI: https://doi.org/10.1007/s00373-007-0743-y