Coloring random graphs online without creating monochromatic subgraphs
Consider the following random process: The vertices of a binomial random graph Gn,p are revealed one by one, and at each step only the edges induced by the already revealed vertices are visible. Our goal is to assign to each vertex one from a fixed number r of available colors immediately and irrevocably without creating a monochromatic copy of some fixed graph F in the process. Our first main result is that for any F and r, the threshold function for this problem is given by p0(F,r,n) = n-1/m*1(F,r), where m*1(F,r) denotes the so-called online vertex-Ramsey density of F and r. This parameter is defined via a purely deterministic two-player game, in which the random process is replaced by an adversary that is subject to certain restrictions inherited from the random setting. Our second main result states that for any F and r, the online vertex-Ramsey density m*1(F,r) is a computable rational number.
random graph ramsey property online process