Erdös-Ko-Rado from intersecting shadows

• Autorzy: Gyula O.H. Katona , Ákos Kisvölcsey
• Języki publikacji: EN

Abstrakty

EN

A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdős-Ko-Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corresponding results.

Słowa kluczowe

EN

Kneser graph; coclique; intersecting family; shadow

Kategorie tematyczne

• 05C35: Extremal problems
• 05D05: Extremal set theory

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2012
• Tom: 32
• Numer: 2
• Strony: 379-382

Daty

wydano

2012

otrzymano

2011-04-28

poprawiono

2011-08-06

zaakceptowano

2011-08-08

Twórcy

autor

Gyula O.H. Katona

• Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1053 Budapest, Reáltanoda u. 13-15, Hungary

autor

Ákos Kisvölcsey

• Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1053 Budapest, Reáltanoda u. 13-15, Hungary

Bibliografia

• [1] P. Borg, A short proof of a cross-interscting theorem of Hilton, Discrete Math. 309 (2009) 4750-4753, doi: 10.1016/j.disc.2008.05.051.
• [2] D.E. Daykin, Erdös-Ko-Rado from Kruskal-Katona, J. Combin. Theory (A) 17 (1974) 254-255, doi: 10.1016/0097-3165(74)90013-2.
• [3] P. Erdös, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math., Oxford 12 (1961) 313-320.
• [4] A.J.W. Hilton, An intersection theorem for a collection of families of subsets of a finite set, J. London Math. Soc. (2) 15 (1977) 369-376, doi: 10.1112/jlms/s2-15.3.369.
• [5] G.O.H. Katona, Intersection theorems for systems of finite sets, Acta Math. Hungar. 15 (1964) 329-337, doi: 10.1007/BF01897141.
• [6] G.O.H. Katona, A theorem of finite sets in: Theory of Graphs, Proc. Colloq. Tihany, 1966, P. Erdös and G.O.H. Katona (Eds.) (Akadémiai Kiadó, 1968) 187-207.
• [7] J.B. Kruskal, The number of simplicies in a complex in: Math. Optimization Techniques, R. Bellman (Ed.) (Univ. of Calif. Press, Berkeley, 1963) 251-278.
• [8] J. Wang and H.J. Zhang, Cross-intersecting families and primitivity of symmetric systems, J. Combin. Theory (A) 118 (2011) 455-462, doi: 10.1016/j.jcta.2010.09.005.
• [9] H.J. Zhang, Primitivity and independent sets in direct products of vertex-transitive graphs, J. Graph Theory 67 (2011) 218-225, doi: 10.1002/jgt.20526.

Identyfikatory

DOI

10.7151/dmgt.1620