Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a graph FF and an integer r≥2r≥2, a partition F̂ of the edge set of FF into at most rr classes, and a graph GG, define cr,F̂(G) as the number of rr-colorings of the edges of GG that do not contain a copy of FF such that the edge partition induced by the coloring is isomorphic to the one of FF.  think of F̂ as the pattern of coloring that should be avoided. The main question is, for a large enough nn, to find the (extremal) graph GG on nn vertices which maximizes cr,F̂(G). This problem generalizes a question of Erdős and Rothschild, who originally asked about the number of colorings not containing a monochromatic clique (which is equivalent to the case where FF is a clique and the partition F̂ contains a single class).  use Holder’s Inequality together with Zykov’s Symmetrization to prove that, for any r≥2r≥2, k≥3k≥3 and any pattern Kk̂ of the clique KkKk, there exists a complete multipartite graph that is extremal. Furthermore, if the pattern Kk̂ has at least two classes, with the possible exception of two very small patterns (on three or four vertices), every extremal graph must be a complete multipartite graph. In the case that r=3r=3 and F̂ is a rainbow triangle (that is, where F=K3F=K3 and each part is a singleton),  show that an extremal graph must be an almost complete graph. Still for r=3r=3,  extend a result about monochromatic patterns of Alon, Balogh, Keevash and Sudakov to some patterns that use two of the three colors, finding the exact extremal graph. For the later two results,  use the Regularity and Stability Method.
['Fabrício Benevides', 'Carlos Hoppen', 'Rudini Menezes Sampaio']
I am writing a research paper on a idea Edge-colorings of graphs avoiding complete graphs with a prescribed coloring
[{"from": "human", "value": "I am writing a research paper on a idea Edge-colorings of graphs avoiding complete graphs with a prescribed coloring"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a graph FF and an integer r\u22652r\u22652, a partition F\u0302 of the edge set of FF into at most rr classes, and a graph GG, define cr,F\u0302(G) as the number of rr-colorings of the edges of GG that do not contain a copy of FF such that the edge partition induced by the coloring is isomorphic to the one of FF.  think of F\u0302 as the pattern of coloring that should be avoided. The main question is, for a large enough nn, to find the (extremal) graph GG on nn vertices which maximizes cr,F\u0302(G). This problem generalizes a question of Erd\u0151s and Rothschild, who originally asked about the number of colorings not containing a monochromatic clique (which is equivalent to the case where FF is a clique and the partition F\u0302 contains a single class).  use Holder\u2019s Inequality together with Zykov\u2019s Symmetrization to prove that, for any r\u22652r\u22652, k\u22653k\u22653 and any pattern Kk\u0302 of the clique KkKk, there exists a complete multipartite graph that is extremal. Furthermore, if the pattern Kk\u0302 has at least two classes, with the possible exception of two very small patterns (on three or four vertices), every extremal graph must be a complete multipartite graph. In the case that r=3r=3 and F\u0302 is a rainbow triangle (that is, where F=K3F=K3 and each part is a singleton),  show that an extremal graph must be an almost complete graph. Still for r=3r=3,  extend a result about monochromatic patterns of Alon, Balogh, Keevash and Sudakov to some patterns that use two of the three colors, finding the exact extremal graph. For the later two results,  use the Regularity and Stability Method."}]