Archive for Mathematical Logic 49 (7-8):799-811 (2010)

Abstract
When no single universal model for a set of structures exists at a given cardinal, then one may ask in which models of set theory does there exist a small family which embeds the rest. We show that for λ+-graphs (λ regular) omitting cliques of some finite or uncountable cardinality, it is consistent that there are small universal families and 2λ > λ+. In particular, we get such a result for triangle-free graphs
Keywords Graphs  Triangle-free graphs  Graphs omitting cliques  Universal models  Infinite combinatorics
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DOI 10.1007/s00153-010-0197-4
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References found in this work BETA

Perfect-Set Forcing for Uncountable Cardinals.Akihiro Kanamori - 1980 - Annals of Mathematical Logic 19 (1-2):97-114.
Perfect Trees and Elementary Embeddings.Sy-David Friedman & Katherine Thompson - 2008 - Journal of Symbolic Logic 73 (3):906-918.
On Universal Graphs Without Instances of CH.Saharon Shelah - 1984 - Annals of Pure and Applied Logic 26 (1):75-87.
Universal Structures in Power ℵ1.Alan H. Mekler - 1990 - Journal of Symbolic Logic 55 (2):466-477.

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