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| taffer... |
 Posted: Sun Nov 01, 2009 2:50 pm |
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If you reply to this thread I assume you have heard about Frankl's
conjecture, or the union closed sets conjecture. But is there an entry
level literature for this conjecture, and related areas of
combinatorics and graph theory? I get the impression it is fairly
isolated. Is the only relevant literature research papers on special
cases of the problem?
For example, I have a couple of books on graph theory that shed some
light on stuff like Hadwiger's conjecture, and the related areas of
combinatorics and graph theory. I understand the union closed sets
problem has connections to finite lattices, but I want to know more
than that. For instance, it would be great to have a graduate or
advanced undergraduate text which touched on the conjecture, because
it is related to what the (hypothetical) book is about.
Thanks |
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| Butch Malahide... |
| Posted: Sun Nov 01, 2009 9:53 pm |
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On Nov 2, 1:38 am, William Elliot <ma... at (no spam) rdrop.remove.com> wrote:
[quote]On Sun, 1 Nov 2009, taffer wrote:
If you reply to this thread I assume you have heard about Frankl's
conjecture, or the union closed sets conjecture.
No I haven't, nor have I found it on the web. What is it?
[/quote]
I bet you tried real hard to find it.
http://en.wikipedia.org/wiki/Union-closed_sets_conjecture |
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| William Elliot... |
| Posted: Mon Nov 02, 2009 2:38 am |
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On Sun, 1 Nov 2009, taffer wrote:
[quote]If you reply to this thread I assume you have heard about Frankl's
conjecture, or the union closed sets conjecture.
[/quote]
No I haven't, nor have I found it on the web. What is it?
Spaces for which the unions of closed sets are closed,
are Alexandroff spaces.
[quote]But is there an entry level literature for this conjecture, and related
areas of combinatorics and graph theory? I get the impression it is
fairly isolated. Is the only relevant literature research papers on
special cases of the problem?
For example, I have a couple of books on graph theory that shed some
light on stuff like Hadwiger's conjecture, and the related areas of
combinatorics and graph theory. I understand the union closed sets
problem has connections to finite lattices, but I want to know more
than that. For instance, it would be great to have a graduate or
[/quote]
up a = { x | a <= x }
Let S be a T0 Alexandroff space. Then x <= y when x in cl {y},
is an (partial) order and B = { up x | x in S } is a base for S.
Conversely, if (S,<=) is an ordered set, then S given the
base { up x | x in S } is a T0 Alexandroff space with the
property that x <= y iff x <= cl {y}.
[quote]advanced undergraduate text which touched on the conjecture, because
it is related to what the (hypothetical) book is about.
[/quote] |
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| taffer... |
| Posted: Wed Nov 04, 2009 3:44 am |
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Guest
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On 2 Nov, 07:38, William Elliot <ma... at (no spam) rdrop.remove.com> wrote:
[quote]On Sun, 1 Nov 2009, taffer wrote:
If you reply to this thread I assume you have heard about Frankl's
conjecture, or the union closed sets conjecture.
No I haven't, nor have I found it on the web. What is it?
Spaces for which the unions of closed sets are closed,
are Alexandroff spaces.
But is there an entry level literature for this conjecture, and related
areas of combinatorics and graph theory? I get the impression it is
fairly isolated. Is the only relevant literature research papers on
special cases of the problem?
For example, I have a couple of books on graph theory that shed some
light on stuff like Hadwiger's conjecture, and the related areas of
combinatorics and graph theory. I understand the union closed sets
problem has connections to finite lattices, but I want to know more
than that. For instance, it would be great to have a graduate or
up a = { x | a <= x }
Let S be a T0 Alexandroff space. Then x <= y when x in cl {y},
is an (partial) order and B = { up x | x in S } is a base for S.
Conversely, if (S,<=) is an ordered set, then S given the
base { up x | x in S } is a T0 Alexandroff space with the
property that x <= y iff x <= cl {y}.
[/quote]
There is the technicality that if the empty set is not present, then
closure under finite unions leads to the empty set being present,
while closure under pairwise unions does not.
Visa vi finite spaces, the topologies on a finite set X are in one to
one correspondence with the reflexive transitive relations on X. Given
a reflexive transitive relation on X, the corresponding topology is T0
if and only if the relation is a partial order.
I think, after searching, Frankl's conjecture is the sort of
conjecture that doesn't really have any surrounding literature that is
in text book form. It does have a very elementary statement, after
all. Anywho, must dash. |
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