×

Ramsey games. (English) Zbl 0551.03034

The Ramsey game R(\(\alpha\),\(\tau\),\(\beta)\) is as follows: two players White and Black alternately pick previously unchosen members of \([\alpha]^{\tau}\). White moves at limit stages and the game ends when \([\alpha]^{\tau}\) is exhausted. White wins if \(\exists A\subset \alpha,\quad tpA=\beta\) with \([A]^{\tau}\subseteq W\) where W is the set of White’s choices; otherwise Black wins. To say a player wins R(\(\alpha\),\(\tau\),\(\beta)\) means he has a winning strategy. The game \(R(\alpha,<\tau,\beta)\) is defined anologously. R(\(\alpha\),\(\tau\),\(\beta)\) and \(R(\alpha,<\tau,\beta)\) are game theoretic versions of the partition relations \(\alpha \to (\beta)_ 2^{\tau}\) and \(\alpha \to (\beta)_ 2^{<\tau}\) respectively.
The authors prove several results about these games, e.g., if \(2^{\kappa}=\kappa^+\) then Black wins \(R(\kappa^+,2,\kappa^+),\) if \(\kappa\) is a strong limit cardinal then White wins R(\(\kappa\),2,\(\kappa)\); if \(\kappa \to (\beta)_ 2^{\omega}\) holds for a limit ordinal \(\beta\) then White wins \(R(\kappa,<\omega,\beta).\) This last result occasions the definition of ”sequoia” which the authors hope will have a life of its own.
Reviewer: J.M.Plotkin

MSC:

03E55 Large cardinals
03E05 Other combinatorial set theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. E. Baumgartner, Results and independence proofs in combinatorial set theory, Doctoral Dissertation, University of California, Berkeley, California, 1970.
[2] J. E. Baumgartner, F. Galvin, R. McKenzie and R. Laver, Game theoretic versions of partition relations, Infinite and Finite Sets, Colloquia Math. Soc. János Bolyai, Vol. 10, Keszthely, Hungary, 1973. · Zbl 0307.90097
[3] Keith J. Devlin, Some weak versions of large cardinal axioms, Ann. Math. Logic 5 (1972 – 1973), 291 – 325. · Zbl 0279.02051
[4] K. J. Devlin and J. B. Paris, More on the free subset problem, Ann. Math. Logic 5 (1972 – 1973), 327 – 336. · Zbl 0279.02052
[5] P. Erdös and R. Rado, Combinatorial theorems on classifications of subsets of a given set, Proc. London Math. Soc. (3) 2 (1952), 417 – 439. · Zbl 0048.28203
[6] A. Hajnal, Proof of a conjecture of S. Ruziewicz, Fund. Math. 50 (1961/1962), 123 – 128. · Zbl 0100.28003
[7] P. Komjath, Preprint.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.