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Idiosynchromatic Poetry
Weinert, Thilo
Advisor : Prof. Dr. Stefan Geschke
Introduction, Classical Results. . .
Ramsey theory is generally concerned with problems of finding structures with some kind of homogeneity in superstructures. Often a structure contains an homogeneous substructure of a certain sort if it is
itself large enough. In some contexts the notion of size can not only be
interpreted as cardinality but also as order-type.
Notation 1 For ordinal numbers α and β we let r(α, β) denote the
least ordinal number γ such that any graph on γ either has an independent set of order-type α or a complete subgraph induced by a set of
order-type β.
The easiest non-trivial result in Ramsey theory probably is the following:
Theorem 2 (Theorem on Friends and Strangers) r(3, 3) = 6.
The name of this originates in a popular interpretation of the theorem
stating that on any party with six guests one can find a group of three
mutually knowing each other or a group of three who are mutually
unacquainted.
Ramsey theory proper started when in 1928 british philosopher, economist and mathematician Frank Plumpton Ramsey proved the following
theorem.
Theorem 3 (Frank Plumpton Ramsey, 1928)
r(ω, ω) = ω.
. . . and an eclectic Definition
General Results
Theorem 5 (Pál Erdős & András Hajnal, 1971)
If the continuum hypothesis
holds true then r(ω1 ω, 3) > ω1 ω.
Notation 10 r(Im , An ) denotes the least natural number k such that
any edge-tricoloured digraph on k vertices either contains an independent m-tuple or a set of n vertices such that all triples within are agreeable.
Theorem 6 (James Earl Baumgartner, 1989)
If the continuum hypothesis fails yet Martin’s Axiom
holds true then r(ω1 ω, n) = ω1 ω for all natural n.
Theorem 11 For natural numbers m, n and for κ = ω or κ a weakly
compact cardinal we have
r(κ2 m, n) = κ2 r(Im , An ).
Notation 7 r(Im , Ln ) denotes the least natural number k such that
any digraph on k vertices either contains an independent m-tuple or a
transitive subtournament of size n.
Theorem 8 (Erdős & Rado, 1956, Baumgartner, 1974)
For natural numbers m, n and for
any infinite cardinal κ we have
r(κm, n) = κr(Im , Ln ).
It turned out to be possible to prove analogues of Theorem 8 using a
somewhat bizarre-looking definition.
Definition 9
A triple is called agreeable if and only if it is one of the following.
Theorem 12 For all m ∈ ω \ 2 we have
r(Im , A3 ) 6
(2m + 1)(m2 + 4m − 6)
.
3
Notation 13 r(Im , Sn ) denotes the least natural number k such that
any edge-bicoloured digraph on k vertices either contains an independent m-tuple or a set of n vertices such that all triples within are strongly agreeable.
Theorem 14 For natural numbers m, n, for a weakly compact cardinal
κ and for an infinite cardinal λ < κ we have
r(κλm, n) = κλr(Im , Sn ).
Theorem 15 If the continuum hypothesis fails yet Martin’s Axiom
holds true then for natural numbers m and n we have
After the publication of their seminal paper [56ER] in which Pál Erdős
and Richard Rado—among other things—proved Theorem 8 for κ = ω,
interest in Ramsey theory increased.
When one considers graphs on countable structures then Shoenfield’s
Theorem tells us that we cannot expect to prove statements to be
independent from ZFC by the method of forcing.
r(ω1 ωm, n) = ω1 ωr(Im , Sn ).
Theorem 16 For all m ∈ ω \ 2 and all n ∈ ω \ 3 we have
n−1 X
1
i + m − 2 i
r(Im , Sn ) 6
3+
4 .
4
i
i=0
Theorem 4 (Ernst Specker, 1957)
r(ω 2 , n) = ω 2 for all natural n.
Corollary 17 For all m ∈ ω \ 2 we have
However if one is concerned with graphs on uncountable structures
then independence of ZFC can come into play.
If an agreeable triple contains no green arrow we call it strongly
agreeable.
r(Im , S3 ) 6 m(2m − 1).
Counterexamples. . .
. . . and a Table
References
Theorem 20 r(I3 , S3 ) = 15.
[12We] Thilo Weinert. Idiosynchromatic Poetry, submitted.
[09GS] Agelos Georgakopoulos and Philipp Sprüssel. On 3-coloured tournaments. Preprint, April 2009, http://www.arxiv.org/pdf/0904.1967.pdf
Theorem 18 r(I2 , S3 ) = 6.
[09Rd] Stanislaw P. Radziszowski, Small Ramsey Numbers, The Electronic
Journal of Combinatorics (2009), DS1.12.
The lower bound is provided by the
counterexample to the right, the upper bound is given by Corollary 17.
[00FS] Harold M. Fredricksen and Melvin M. Sweet, Symmetric Sum-Free
Partitions and Lower Bounds for Schur Numbers, Electronic Journal of
Combinatorics, http://www.combinatorics.org/, #R32, 7 (2000), 9 pages.
Theorem 19 r(I2 , A3 ) = 10.
[97MR] Brendan Damien McKay and Stanislaw P. Radziszowski. Subgraph Counting Identities and Ramsey Numbers, Journal of Combinatorial Theory, Series B, 69(2):193-209,1997.
http://dx.doi.org/10.1006/jctb.1996.1741.
[97LM] Jean Ann Larson and William J. Mitchell. On a problem
of Erdős and Rado. Annals of Combinatorics, 1(3):245-252, 1997.
http://dx.doi.org/10.1007/BF02558478.
[89Ba] James Earl Baumgartner. Remarks on partition ordinals. in Set Theory and its applications(Toronto, ON, 1987), volume 1401 of Lecture
Notes in Mathematics, pages 5–17. Springer, Berlin, 1989.
[82HI] Raymond Hill and Robert Wylie Irving, On group partitions associated with lower bounds for symmetric Ramsey numbers, European Journal
of Combinatorics vol. 3 (1982), pp. 35-50.
Algebraically this can be
seen as the tournament on
Z9 where from every i ∈ Z9
there is a green arrow pointing at i + 1, blue arrows
pointing both at i + 2 and
i + 3 and a red arrow pointing at i + 4.
This is algebraically interpretable as the tournament on Z3 × Z3 where
from every hi, ji there is
a green arrow pointing at
hi + 1, ji and blue arrows
pointing at hi, j + 1i and
hi+1, j+1i and hi+2, j+1i.
The counterexamples above yield the lower bound, the upper bound is
given by Theorem 12.
[79No] Eva Nosal. Partition relations for denumerable ordinals. Journal
of
Combinatorial
Theory
(B), 27(2):190–197, 1979.
http://dx.doi.org/10.1016/0095-8956(79)90080-7.
[74Ba] James Earl Baumgartner. Improvement of a partition theorem of Erdős
and Rado. Journal of Combinatorial Theory Series A, 17:134–137,
1974.
[74Be] Jean-Claude Bermond. Some Ramsey numbers for directed graphs. Discrete Mathematics, 9:313–321, 1974.
The lower bound is given by the counterexample above. It can be viewed as a digraph on Z14 where v 7→r v + 1, v 7→b v + 2, v 7→b v − 3 and
v 7→r v − 4 for all v ∈ Z14 .
The upper bound is established by Corollary 17 above.
3
4
5
6
7
8
9
10 + n
3
6
9
14
18
23
28
36
4
9
18
25
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ω3
ω4
ω4
ω5
ω5
ω5
ω5
ω6
ω 1+2dld(10+n)e
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ω 10
ω 10
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κλ2
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[74No] Eva Nosal, On a partition relation for ordinal numbers. The Journal of
the London Mathematical Society (2), 8:306–310, 1974.
[71EH] Pál Erdős and András Hajnal. Ordinary partition relations for ordinal
numbers. Periodica Mathematica Hungarica, 1(3):171–185, 1971.
[70RP] Kenneth Brooks Reid, Junior and Ernest Tilden Parker. Disproof
of a conjecture of Erdős and Moser on tournaments. Journal of Combinatorial Theory, 9:225–238, 1970.
[64EM] Pál Erdős and Leo Moser. On the representation of directed graphs
as unions of orderings. Magyar Tud. Akad. Mat. Kutató Int. Közl,
9:125–132, 1964.
[57Sp] Ernst Specker. Teilmengen von Mengen mit Relationen. Commentationes Mathematicae Helvetia, 31:302–314, 1957.
[56ER] Pál Erdős and Richard Rado. A partition calculus in set theory. Bulletin of the American Mathematical Society 62:427–489, 1956.
[55GG] Robert Ewing Greenwood junior and Andrew Mattei Gleason,
Combinatorial Relations and Chromatic Graphs, Canadian Journal of
Mathematics, 7:1–7, 1955.
[30Rm] Frank Plumpton Ramsey. On a problem in formal logic. Proceedings
of the London Mathematical Society 30:264–286, 1930.
2012 Hausdorff Centre for Mathematics
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