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Universality among graphs omitting a complete bipartite graph

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Abstract

For cardinals λ,κ,θ we consider the class of graphs of cardinality λ which has no subgraph which is (κ,θ)-complete bipartite graph. The question is whether in such a class there is a universal one under (weak) embedding. We solve this problem completely under GCH. Under various assumptions mostly related to cardinal arithmetic we prove non-existence of universals for this problem. We also look at combinatorial properties useful for those problems concerning κ-dense families.

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Authors and Affiliations

  1. Einstein Institute of Mathematics Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Saharon Shelah

  2. Department of Mathematics Hill Center — Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Saharon Shelah

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  1. Saharon Shelah
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Correspondence to Saharon Shelah.

Additional information

The author thanks Alice Leonhardt for the beautiful typing. This research was supported by the United States-Israel Binational Science Foundation. Publication 706.

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Shelah, S. Universality among graphs omitting a complete bipartite graph. Combinatorica 32, 325–362 (2012). https://doi.org/10.1007/s00493-012-2033-4

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  • DOI: https://doi.org/10.1007/s00493-012-2033-4

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