Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider vectors from $\{0,1\}^n$. The ight of such a vector $v$ is the sum of the coordinates of $v$. The distance ratio of a set $L$ of vectors is ${\rm dr}(L):=\max \{\rho(x,y):\ x,y \in L\}/ \min \{\rho(x,y):\ x,y \in L,\ x\neq y\},$ where $\rho(x,y)$ is the Hamming distance beten $x$ and $y$.  prove that (a) for every constant $\lambda>1$ there are no positive constants $\alpha$ and $C$ such that every set $K$ of at least $\lambda^p$ vectors with ight $p$ contains a subset $K'$ with $|K'|\ge |K|^{\alpha}$ and ${\rm dr}(K')\le C$, % even when $|K|\ge \lambda$, #R##N#(b) For a set $K$ of vectors with ight $p$, and a constant $C>2$, there exists $K'\subseteq K$ such that ${\rm dr}(K')\le C$ and $|K'| \ge |K|^\alpha$, where $\alpha = 1/ \lceil \log(p/2)/\log(C/2) \rceil$.
['Gregory Gutin', 'Mark Jones']
I am writing a research paper on a idea Note on Existence and Non-Existence of Large Subsets of Binary Vectors with Similar Distances
[{"from": "human", "value": "I am writing a research paper on a idea Note on Existence and Non-Existence of Large Subsets of Binary Vectors with Similar Distances"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider vectors from $\\{0,1\\}^n$. The ight of such a vector $v$ is the sum of the coordinates of $v$. The distance ratio of a set $L$ of vectors is ${\\rm dr}(L):=\\max \\{\\rho(x,y):\\ x,y \\in L\\}/ \\min \\{\\rho(x,y):\\ x,y \\in L,\\ x\\neq y\\},$ where $\\rho(x,y)$ is the Hamming distance beten $x$ and $y$.  prove that (a) for every constant $\\lambda>1$ there are no positive constants $\\alpha$ and $C$ such that every set $K$ of at least $\\lambda^p$ vectors with ight $p$ contains a subset $K'$ with $|K'|\\ge |K|^{\\alpha}$ and ${\\rm dr}(K')\\le C$, % even when $|K|\\ge \\lambda$, #R##N#(b) For a set $K$ of vectors with ight $p$, and a constant $C>2$, there exists $K'\\subseteq K$ such that ${\\rm dr}(K')\\le C$ and $|K'| \\ge |K|^\\alpha$, where $\\alpha = 1/ \\lceil \\log(p/2)/\\log(C/2) \\rceil$."}]