Supported by CNPq (Partially; Proc. 308509/2007-2 and Proc. 484154/2010-9); CAPES-DAAD Collaboration
Upper bounds on probability thresholds for asymmetric Ramsey properties
Article first published online: 18 JUL 2012
Copyright © 2012 Wiley Periodicals, Inc.
Random Structures & Algorithms
Volume 44, Issue 1, pages 1–28, January 2014
How to Cite
Kohayakawa, Y., Schacht, M. and Spöhel, R. (2014), Upper bounds on probability thresholds for asymmetric Ramsey properties. Random Struct. Alg., 44: 1–28. doi: 10.1002/rsa.20446
Supported by Heisenberg-Programme of the Deutsche Forschungsgemeinschaft (DFG Grant SCHA 1263/4-1); CAPES-DAAD Collaboration.
Supported by Swiss National Science Foundation.
- Issue published online: 25 NOV 2013
- Article first published online: 18 JUL 2012
- Manuscript Accepted: 21 MAR 2012
- Manuscript Received: 2 MAY 2011
- Ramsey theory;
- random graphs;
- threshold functions;
- non-diagonal Ramsey properties
Given two graphs G and H, we investigate for which functions the random graph (the binomial random graph on n vertices with edge probability p) satisfies with probability 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