## Two cardinal versions of diamond.(English)Zbl 0798.03047

Roughly speaking, $$\diamondsuit_{\kappa,\lambda}$$ asserts the existence of a sequence of size $$<\kappa$$ sets that capture every subset of $$\lambda$$ on a stationary set. The paper is devoted to the study of $$\diamondsuit_{\kappa, \lambda}$$ and related principles, which are for instance obtained by considering sequences of larger sets, or by requesting the simultaneous capture of many subsets of $$\lambda$$. Our main result is that $$\diamondsuit_{\kappa, \lambda}$$ holds in case $$\lambda> 2^{< \kappa}$$.
Reviewer: P.Matet (Caen)

### MSC:

 300000 Other combinatorial set theory 3e+35 Consistency and independence results

### Keywords:

diamond; club filter; stationary set
Full Text:

### References:

 [1] J. E. Baumgartner and A. D. Taylor,Saturation properties of ideals in generic extensions. I, Trans. Am. Math. Soc.270 (1982), 557–574. · Zbl 0485.03022 [2] J. E. Baumgartner,On the size of closed unbounded sets, Ann. Pure Appl. Logic54 (1991), 195–227. · Zbl 0746.03040 [3] M. R. Burke and M. Magidor,Shelah’s pcf theory and its applications, Ann. Pure Appl. Logic50 (1990), 207–254. · Zbl 0713.03024 [4] K. J. Devlin,Some weak versions of large cardinal axioms, Ann. Math. Logic5 (1973), 291–325. · Zbl 0279.02051 [5] C. A. Di Prisco and W. Marek,Some aspects of the theory of large cardinals, inMathematical Logic and Formal Systems (L. P. de Alcantara, ed.), Lecture Notes in Pure and Applied Mathematics 94, Dekker, New York, 1985, pp. 87–139. [6] Q. Feng,On weakly stationary sets, Proc. Am. Math. Soc.105 (1989), 727–735. · Zbl 0663.03042 [7] M. Gitik, Nonsplitting subset of $$\mathcal{P}_k \left( {k^ + } \right),$$ J. Symbolic Logic50 (1985), 881–894. · Zbl 0601.03021 [8] J. Gregory,Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic41 (1976), 663–671. · Zbl 0347.02044 [9] T. J. Jech,Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic5 (1973), 165–198. · Zbl 0262.02062 [10] R. B. Jensen,The fine structure of the constructible hierarchy, Ann. Math. Logic4 (1972), 229–308. · Zbl 0257.02035 [11] D. Kueker,Countable approximations and Löwenheim-Skolem theorems, Ann. Math. Logic11 (1977), 57–103. · Zbl 0364.02009 [12] K. Kunen,Set Theory, North-Holland, Amsterdam, 1980. [13] M. Magidor,Representing sets of ordinals as countable unions of sets in the core model, Trans. Am. Math. Soc.317 (1990), 91–126. · Zbl 0714.03045 [14] P. Matet,Partitions and Diamond, Proc. Am. Math. Soc.97 (1986), 133–135. [15] P. Matet,On diamond sequences, Fund. Math.131 (1988), 35–44. · Zbl 0663.03035 [16] P. Matet, Concerning stationary subsets of $$\left[ \lambda \right]^{< k} ,$$ inSet Theory and its Applications (J. Steprāns and S. Watson, eds.), Lecture Notes in Mathematics1401, Springer, Berlin, 1989, pp. 119–127. [17] T. K. Menas,On strong compactness and supercompactness, Ann. Math. Logic7 (1974), 327–359. · Zbl 0299.02084 [18] S. Shelah,On successors of singular cardinals, inLogic Colloquium 78 (M. Boffa, D. van Dalen and K. Mc Aloon, eds.), North-Holland, Amsterdam, 1979, pp. 357–380. [19] S. Shelah,Proper Forcing, Lecture Notes in Mathematics940, Springer, Berlin, 1982. [20] S. Shelah,More on stationary coding, inAround Classification Theory of Models, Lecture Notes in Mathematics1182, Springer, Berlin, 1986, pp. 224–246. [21] S. Shelah,Cardinal arithmetic for skeptics, Bull. Am. Math. Soc.26 (1992), 197–210. · Zbl 0771.03017 [22] C. I. Steinhorn and J. H. King, The uniformization property for 2, Isr. J. Math.36 (1980), 248–256. · Zbl 0451.03018 [23] N. H. Williams,Combinatorial Set Theory, Studies in Logic and the Foundations of Mathematics 91, North-Holland, Amsterdam, 1977.
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