Abstract
By considering the number of maximal chains going through each element of an arbitrary poset, we prove an extension of Erdős’s generalisation of Sperner’s Theorem, together with a partial converse. By considering the number of maximal chains between pairs of comparable elements, we also prove a generalisation of the LYM inequality.
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References
Anderson, I.: Combinatorics of Finite Sets. Clarendon Press, Oxford (1987)
Beck, M., Zaslavsky, T., Shorter, A: Simpler, stronger proof of the Meshalkin-Hochberg-Hirsch bounds on componentwise antichains. Journal of Combinatorial Theory, Series A 100, 196–199 (2002)
Beck, M., Wang, X., Zaslavsky, T.: A Unifying Generalization of Sperner’s Theorem. In: Gyõri, E., Katona, G.O.H., Lovasz, L. (eds.) More Sets, Graphs and Numbers: a Salute to Vera Sos and Andras Hajnal, In: Bolyai Soc. Math. Stud., vol. 15, pp 9–24. Springer, Janos Bolyai Mathematical Society, Berlin, Budapest (2006)
Bollobäs, B.: On generalized graphs. Acta Math. Acad. Sci. Hung. 16, 447–452 (1965)
Chudak, F., Griggs, J.R.: A new extension of Lubell’s inequality to the lattice of divisors. Stud. Sci. Math. Hung. 35, 347–351 (1999)
Engel, K.: Sperner Theory, Encyclopedia of Mathematics and its Applications, vol. 65. Cambridge University Press, Cambridge (1997)
Erdős, P.: On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. 51, 898–902 (1945)
Erdös, P.L., Katona, G.O.H.: Convex Hulls of more–Part Sperner Families. Graph and Combinatorics 2, 123–134 (1986)
Grätzer, G.: General Lattice Theory, 2nd edn. Basel Switzerland, Birkhäuser Verlag (1998)
Hochberg, M., Hirsch, W.M., families, Sperner: Sperner families, s-systems, and a theorem of Meshalkin. Ann. New York Acad. Sci. 175, 224–237 (1970)
Kleitman, D.J.: On an extremal property of antichains in partial orders: the LYM property and some of its implications and applications, Combinatorics (M. Hall and J. H. van Lint, editors). Math. Centre Tracts, Amsterdam 55, 77–90 (1974)
Lubell, D.: A short proof of Sperner’s lemma. J. Combin. Theory 1, 299 (1966)
Meshalkin, L.D.: Generalization of Sperner’s Theorem on the number of subsets of a finite set. Teor. Verojatnost. i Primenen 8, 219–220 (1963). English transl.: Theor. Probab. Appl., 8 (1963), 203–204.[In Russian.]
Harper, L.H., Rota, G.-C.: Matching theory, an introduction. In: Ed, P. (ed.) Advances in Probability and Related Topics, vol. 1, pp 169–215. Marcel Dekker, New York (1971)
Sperner, E.: Ein Satz ü,ber Untermengen einer endlichen Menge. Math. Z. 27, 544–548 (1928)
Yamamoto, K.: Logarithmic order of free distributive lattices. J. Math. Soc. Japan 6, 343–353 (1954)
Acknowledgments
This research was done when the first author was a postdoctoral researcher at IPM (Institute for Research in Fundamental Sciences). He was also supported in part by INSF (Iran National Science Foundation). We would like to thank the anonymous referee, who helped us improve the clarity and accessibility of this paper.
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Borujeni, S.H.A., Bowler, N. Investigating posets via their maximal chains. Order 37, 299–309 (2020). https://doi.org/10.1007/s11083-019-09506-2
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DOI: https://doi.org/10.1007/s11083-019-09506-2