Universality among graphs omitting a complete bipartite graph |
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| Authors: | Saharon Shelah |
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| Institution: | 1. Einstein Institute of Mathematics Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
2. Department of Mathematics Hill Center — Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA
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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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