A powerful tool of extremal set theory, the cycle method is surveyed in the paper. It works, however only when the non-emptyness of the pairwise intersections of the members of the family is assumed. If these intersections have to be at least 2, the method fails: the celebrated Complete Intersection Theorem by Ahlswede and Khachatrian cannot be proved by this method. We show the reasons and some attempts to overcome the difficulties.
KeywordsConvex Hull Cyclic Permutation Steiner System Cycle Method Profile Vector
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- K. Engel, Sperner Theory, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1997.Google Scholar
- K. Engel, Péter L. Erdös, “Sperner families satisfying additional conditions and their convex hulls”, Graphs and Combinatorics, 5, 1988, 50–59.Google Scholar
- P. Frankl, Z. Füredi, “Families of finite sets with a missing intersection”, Finite and Infinite Sets (Proc. 6th Hungar. Colloq. on Combinatorics, Eger, 1981), Eds. A. Hajnal, L. Lovâsz and V.T. Sós, vol. 37, North Holland, Amsterdam, 1984, 305–318.Google Scholar
- Z. Füredi, personal communication.Google Scholar
- Z. Füredi, D. Kleitman, “The minimal number of zero sums”, in Cornbinatorics, Paul Erdós is eighty,Vol. I, pp; 159–172, Keszthely, Hungary, 1993, D. Miklós et al., Eds., Bolyai Society Mathematical Studies 1(1993),Budapest, Hungary.Google Scholar
- R. Howard, Gy. Károlyi, L.A. Székely, “Towards a Katona type proof for the 2-intersecting Erdós-Ko-Rado theorem”, preprint.Google Scholar
- G.O.H. Katona, “Extremal problems for hypergraphs”, Combinatorics, Ed. by M. Hall, Jr., J.H. van Lint, D. R,eidel, Dordrecht/Boston, 1975, 215–244.Google Scholar
- G.O.H. Katona, G. Schild, “Linear inequalities describing the class of Sperner families of subsets I”, Topics in Combinatorics and Graph Theory (Essays in Honour of Gerhard Ringel), Ed. R. Bodendiek and R. Henn, Physica-Verlag, Heidelberg, 1990, 413–420.Google Scholar