Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, December 03, 2009, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Reto Spöhel

Upper bounds for asymmetric Ramsey properties of random graphs

Consider the following problem: Is there a coloring of the edges of the random graph $G_{n,p}$ with two colors such that there is no monochromatic copy of some fixed graph $F$? A celebrated result by Rödl and Rucinski (1995) states a general threshold function $p_0(F,n)$ for the existence of such a coloring. Kohayakawa and Kreuter (1997) conjectured a general threshold function for the asymmetric case (where different graphs $F_1$ and $F_2$ are forbidden in the two colors), and verified this conjecture for the case where both graphs are cycles.

Implicit in their work is the following more general statement: The conjectured threshold function is an upper bound on the actual threshold provided that i) the two graphs satisfy some balancedness condition, and ii) the so-called K{\L}R-Conjecture is true for the sparser of the two graphs. We present a new upper bound proof that does not depend on the K{\L}R-Conjecture. Together with earlier lower bound results [Marciniszyn, Skokan, S., Steger (2006)], this yields in particular a full proof of the Kohayakawa-Kreuter conjecture for the case where both graphs are cliques.

Joint work with Yoshiharu Kohayakawa and Mathias Schacht.

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