| author: | Richard P. Anstee and Peter Keevash |
|---|---|
| title: | Pairwise Intersections and Forbidden Configurations |
| keywords: | forbidden configurations, extremal set theory, intersecting set systems, uniform set systems, (0,1)-matrices |
| abstract: |
Let
f
denote the maximum size of a family
m
(a,b,c,d)
F
m
-element set for which there is no pair of subsets
A,B∈
with
F
|A ∩B|≥a
,
|
,
A
∩B|≥b
|A ∩
, and
B
| ≥c
|
.
A
∩
B
|≥d
By symmetry we can assume
a ≥d
and
b ≥c
. We show that
f
is
m
(a,b,c,d)
Θ(m
if either
a+b-1
)
b>c
or
a,b≥1
. We also show that
f
is
m
(0,b,b,0)
Θ(m
and
b
)
f
is
m
(a,0,0,d)
Θ(m
. This can be viewed as a result concerning forbidden
configurations and is further evidence for a conjecture of
Anstee and Sali. Our key tool is a strong stability version
of the Complete Intersection Theorem of Ahlswede and
Khachatrian, which is of independent interest.
a
)
|
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| reference: | Richard P. Anstee and Peter Keevash (2005), Pairwise Intersections and Forbidden Configurations, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 17-20 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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| ps-source: | dmAE0104.ps (126 K) |
| pdf-source: | dmAE0104.pdf (127 K) |
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