Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, June 01, 2006, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Reto Spöhel

Online Ramsey Games in Random Graphs

Consider the following one-player game: The vertices of a random graph are revealed to the player one by one, in a random order, along with all edges induced by the vertices revealed so far. The player has to assign one of $r$ available colors to each vertex immediately, without creating a monochromatic copy of some fixed graph $F$. For which values of $p$ can the player asymptotically almost surely (a.a.s) color the entire random graph $G_{n,p}$? We say that $p_0(n)$ is a threshold for this game if there is a strategy such that the player a.a.s.~succeeds if $p\ll p_0$, but a.a.s.~fails with any strategy if $p\gg p_0$.

We prove explicit thresholds $p_0(F,r,n)$ for a large family of graphs $F$ including cliques and cycles of arbitrary size, and an arbitrary number $r$ of colors. In particular, we show that the order of magnitude of the threshold depends on the number of colors, in contrast to the offline case.

Joint work with Martin Marciniszyn and Angelika Steger.

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