Universality among graphs omitting a complete bipartite graph

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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Correspondence to Saharon Shelah.

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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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Mathematics Subject Classification (2010)

  • 03E05
  • 05C63
  • 03E04