## Journal of Symbolic Logic

### Universal graphs at the successor of a singular cardinal

#### Abstract

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality \aleph0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are μ++ graphs on μ+ that taken jointly are universal for the graphs on μ+, while $2μ+ \gg μ++$. The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal κ exists. The result on the existence of universal graphs is obtained as a specific application of a more general method.

#### Article information

Source
J. Symbolic Logic, Volume 68, Issue 2 (2003), 366- 388.

Dates
First available in Project Euclid: 11 May 2003

https://projecteuclid.org/euclid.jsl/1052669056

Digital Object Identifier
doi:10.2178/jsl/1052669056

Mathematical Reviews number (MathSciNet)
MR1976583

Zentralblatt MATH identifier
1055.03030

#### Citation

Džamonja, Mirna; Shelah, Saharon. Universal graphs at the successor of a singular cardinal. J. Symbolic Logic 68 (2003), no. 2, 366-- 388. doi:10.2178/jsl/1052669056. https://projecteuclid.org/euclid.jsl/1052669056

#### References

• C. C. Chang and H. J. Keisler Model theory, 1 ed., North-Holland,1973, latest edition 1990.
• Z. Füredi and P. Komjath Nonexistence of universal graphs without some trees, Combinatorica, vol. 17 (1997), pp. 163--171.
• R. Grossberg and S. Shelah On universal locally finite groups, Israel Journal of Mathematics, vol. 44 (1983), pp. 289--302.
• M. Džamonja and S. Shelah On the existence of universals, submitted.
• M. Gitik and S. Shelah On densities of box products, Topology and its Applications, vol. 88 (1998), no. 3, pp. 219--237.
• A. Kanamori, W. Reindhart, and R. Solovay Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73--116.
• M. Kojman Representing embeddability as set inclusion, Journal of LMS (2nd series),(1998), no. 158, (58) (2), pp. 257--270.
• P. Komjath and S.Shelah Universal graphs without large cliques, Journal of Combinatorial Theory (Series B),(1995), pp. 125--135.
• M. Kojman and S. Shelah Non-existence of universal orders in many cardinals, Journal of Symbolic Logic, vol. 57 (1992), pp. 875--891.
• R. Laver Making the supercompactness of $\kappa$ indestructible under $\kappa$-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), no. 4, pp. 385--388.
• M. Magidor On the singular cardinals problem I, Israel Journal of Mathematics, vol. 28 (1977), pp. 1--31.
• A. Mekler and S. Shelah Uniformization principles, Journal of Symbolic Logic, vol. 54 (1989), no. 2, pp. 441--459.
• L. Radin Adding closed cofinal sequences to large cardinals, Annals of Mathematical Logic, vol. 22 (1982), pp. 243--261.
• R. Rado Universal graphs and universal functions, \looseness=1 Acta Arithmetica, vol. 9 (1964), pp. 331--340.
• S. Shelah Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer,1998.