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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 ω ω ω ω ω ω ω ω ω ω2 ω4 ω8 ω14 ω28 ω3 ω9 2 ω ω2 ω2 ω2 ω2 ω2 ω2 ω2 ω2 2 2 ω 2 ω 10 ω3 ω4 ω4 ω5 ω5 ω5 ω5 ω6 ω 1+2dld(10+n)e ω4 ω7 ω7 ω 10 ω 10 ω 10 ω 10 5+m 9+2m 9+2m 13+3m 13+3m 13+3m 13+3m 17+4m 1+(4+m)dld(10+n)e ω ω ω ω ω ω ω ω ω ωω ωω ωω ωω ωω ωω ωω ωω ωω 2 2 2 ωω ωω ωω κλ2 κλ6 κλ3 κλ15 [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 [email protected]