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Upper bounds on probability thresholds for asymmetric Ramsey properties

Authors


  • Supported by CNPq (Partially; Proc. 308509/2007-2 and Proc. 484154/2010-9); CAPES-DAAD Collaboration

  • Supported by Heisenberg-Programme of the Deutsche Forschungsgemeinschaft (DFG Grant SCHA 1263/4-1); CAPES-DAAD Collaboration.

  • Supported by Swiss National Science Foundation.

Abstract

Given two graphs G and H, we investigate for which functions math formula the random graph math formula (the binomial random graph on n vertices with edge probability p) satisfies with probability math formula that every red-blue-coloring of its edges contains a red copy of G or a blue copy of H. We prove a general upper bound on the threshold for this property under the assumption that the denser of the two graphs satisfies a certain balancedness condition. Our result partially confirms a conjecture by the first author and Kreuter, and together with earlier lower bound results establishes the exact order of magnitude of the threshold for the case in which G and H are complete graphs of arbitrary size. In our proof we present an alternative to the so-called deletion method, which was introduced by Rödl and Ruciński in their study of symmetric Ramsey properties of random graphs (i.e. the case G = H), and has been used in many proofs of similar results since then.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 1–28, 2014

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