Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph $$H$$H is its extremal density, as an induced subgraph of $$G$$G, where $$|G| \rightarrow \infty $$|G|??. Already for $$4$$4-vertex graphs many questions are still open. Thus, the inducibility of the $$4$$4-path was addressed in a construction of Exoo (Ars Combin 22:5---10, 1986), but remains unknown. Refuting a conjecture of Erd?s, Thomason (Combinatorica 17(1):125---134, 1997) constructed graphs with a small density of both $$4$$4-cliques and $$4$$4-anticliques. In this note,  merge these two approaches and construct better graphs for both problems.
['Chaim Even-Zohar', 'Nati Linial']
I am writing a research paper on a idea A Note on the Inducibility of $$4$$4-Vertex Graphs
[{"from": "human", "value": "I am writing a research paper on a idea A Note on the Inducibility of $$4$$4-Vertex Graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph $$H$$H is its extremal density, as an induced subgraph of $$G$$G, where $$|G| \\rightarrow \\infty $$|G|??. Already for $$4$$4-vertex graphs many questions are still open. Thus, the inducibility of the $$4$$4-path was addressed in a construction of Exoo (Ars Combin 22:5---10, 1986), but remains unknown. Refuting a conjecture of Erd?s, Thomason (Combinatorica 17(1):125---134, 1997) constructed graphs with a small density of both $$4$$4-cliques and $$4$$4-anticliques. In this note,  merge these two approaches and construct better graphs for both problems."}]