Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T14:46:23.489Z Has data issue: false hasContentIssue false
  • English
  • Français

On r-Cross Intersecting Families of Sets

Published online by Cambridge University Press:  18 August 2011

PETER FRANKL  and
NORIHIDE TOKUSHIGE
PETER FRANKL
Affiliation:
3-12-25 Shibuya, Shibuya-ku, Tokyo 150-0002, Japan (e-mail: [email protected])
NORIHIDE TOKUSHIGE
Affiliation:
College of Education, Ryukyu University, Nishihara, Okinawa 903-0213, Japan (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (r−1)n ≥ rk and let . Suppose that F1 ∩ ⋅⋅⋅ ∩ Fr ≠ ∅ holds for all Fi ∈ i, 1 ≤ i ≤ r. Then we show that .

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 5 , September 2011 , pp. 749 - 752
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bey, C. (2005) On cross-intersecting families of sets. Graphs Combin. 21 161–168.CrossRefGoogle Scholar
[2]Bollobás, B. (1965) On generalized graphs. Acta Math. Acad. Sci. Hungar 16 447–452.CrossRefGoogle Scholar
[3]Chowdhury, A. and Patkós, B. (2010) Shadows and intersections in vector spaces. J. Combin. Theory Ser. A 117 1095–1106.CrossRefGoogle Scholar
[4]Erdős, P., Ko, C. and Rado, R. (1961) Intersection theorems for systems of finite sets. Quart. J. Math. Oxford (2) 12 313–320.CrossRefGoogle Scholar
[5]Frankl, P. (1976) On Sperner families satisfying an additional condition. J. Combin. Theory Ser. A 20 1–11.CrossRefGoogle Scholar
[6]Frankl, P. (1991) Multiply-intersecting families. J. Combin. Theory Ser. B 53 195–234.CrossRefGoogle Scholar
[7]Gronau, H.-D. O. F. (1982) On Sperner families in which no k sets have an empty intersection III. Combinatorica 2 25–36.CrossRefGoogle Scholar
[8]Matsumoto, M. and Tokushige, N. (1989) The exact bound in the Erdős–Ko–Rado theorem for cross-intersecting families. J. Combin. Theory Ser. A 52 90–97.CrossRefGoogle Scholar
[9]Mubayi, D. and Verstraëte, J. (2005) Proof of a conjecture of Erdős on triangles in set-systems. Combinatorica 25 599–614.CrossRefGoogle Scholar
[10]Pyber, L. (1986) A new generalization of the Erdős–Ko–Rado theorem. J. Combin. Theory Ser. A 43 85–90.CrossRefGoogle Scholar