Abstract
Let X be a finite set of n elements. We say that the family A = {A 1,..., A m } of its distinct subsets is intersecting if A i ∩ A j ≠ Ø holds for any 1 ≤ i ≤ j ≤ m. A classic theorem of Erdós Ko and Rado [2] states that an intersecting family A consisting of k-element subsets, where k ≤ n/2, has at most (\(\left( \begin{matrix} n-1 \\ k-1 \\ \end{matrix} \right)\)) members.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bollobás, B., Sperner systems consisting of pairs of complementary subsets, J. Comb. Theory, Ser. B 15 (1973), 363–366.
Erdós, P., Chao Ko and Rado,. Intersection theorems for systems of finite sets, Q. J. Math. Oxf. II Ser. 12 (1961), 313–318.
Erdós, P.L., Frankl, P. and Katona, G.O.H., Intersecting Sperner families and their convex hulls, Combinatoriea 4 (1984), 21–34.
Fulkerson, D.R., Blocking and anti-blocking pairs of polyhedra, Math. Program. 1 (1971), 168–194.
Fulkerson, D.R., Anti-blocking polyhedra, J. Comb. Theory Ser. B 12 (1972), 50–71.
Greene, C., Katona, G.O.H. and Kleitman, D.J., Extensions of the Erdós-Ko-Rado theorem, SIAM Stud. Appl. Math. 55 (1976), 1–8.
Milner, E.C., A combinatorial theorem on systems of sets, J. London Math. Soc. 43 (1968), 204–206.
Schrijver, A., Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Ltd, Chichester, 1986.
Sperner, E., Ein Satz fiber Untermenge einer endlichen Menge, Math. Z. 27 (1928), 544–548.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Physica-Verlag Heidelberg
About this chapter
Cite this chapter
Katona, G.O.H., Schild, G. (1990). Linear Inequalities Describing the Class of Intersecting Sperner Families of Subsets, I. In: Bodendiek, R., Henn, R. (eds) Topics in Combinatorics and Graph Theory. Physica-Verlag HD. https://doi.org/10.1007/978-3-642-46908-4_47
Download citation
DOI: https://doi.org/10.1007/978-3-642-46908-4_47
Publisher Name: Physica-Verlag HD
Print ISBN: 978-3-642-46910-7
Online ISBN: 978-3-642-46908-4
eBook Packages: Springer Book Archive