Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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 p0F,r,n = n-1/m*1F,r, where m*1F,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*1F,r is a computable rational number.
['Torsten Mütze', 'Thomas Rast', 'Reto Spöhel']
I am writing a research paper on a idea Coloring random graphs online without creating monochromatic subgraphs
[{"from": "human", "value": "I am writing a research paper on a idea Coloring random graphs online without creating monochromatic subgraphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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 p0F,r,n = n-1/m*1F,r, where m*1F,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*1F,r is a computable rational number."}]