Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A family $\mathcal{F}$ of permutations of the vertices of a hypergraph $H$ is called 'pairwise suitable' for $H$ if, for every pair of disjoint edges in $H$, there exists a permutation in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for $H$ is called the 'separation dimension' of $H$ and is denoted by $\pi(H)$. Equivalently, $\pi(H)$ is the smallest natural number $k$ so that the vertices of $H$ can be embedded in $\mathbb{R}^k$ such that any two disjoint edges of $H$ can be separated by a hyperplane normal to one of the axes.  show that the separation dimension of a hypergraph $H$ is equal to the 'boxicity' of the line graph of $H$. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.
['Manu Basavaraju', 'L. Sunil Chandran', 'Martin Charles Golumbic', 'Rogers Mathew', 'Deepak Rajendraprasad']
I am writing a research paper on a idea Boxicity and separation dimension
[{"from": "human", "value": "I am writing a research paper on a idea Boxicity and separation dimension"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A family $\\mathcal{F}$ of permutations of the vertices of a hypergraph $H$ is called 'pairwise suitable' for $H$ if, for every pair of disjoint edges in $H$, there exists a permutation in $\\mathcal{F}$ in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for $H$ is called the 'separation dimension' of $H$ and is denoted by $\\pi(H)$. Equivalently, $\\pi(H)$ is the smallest natural number $k$ so that the vertices of $H$ can be embedded in $\\mathbb{R}^k$ such that any two disjoint edges of $H$ can be separated by a hyperplane normal to one of the axes.  show that the separation dimension of a hypergraph $H$ is equal to the 'boxicity' of the line graph of $H$. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension."}]