相似距离二元向量大集存在与不存在的注记
Note on Existence and Non-Existence of Large Subsets of Binary Vectors with Similar Distances
摘要：
我们考虑来自$\{0,1\}^n$的向量。这样的向量$v$的权重是$v$的坐标之和。矢量集合$L$的距离比是${\rm dr}(L):=\max \{ρ(x,y):\ x,y \in L\}/ \min \{ρ(x,y):\ x,y \in L,\ x\neq y\},$，其中$ρ(x,y)$是$x$和$y$之间的汉明距离。我们证明了：（a）对于每个常数$λ>1$，不存在正的常数$α$和$C$，使得至少具有权$p$的$λ^p$向量的每个集合$K$包含具有$|K'|\ge |K|^α$和${\rm dr}(K')\le C$的子集$K'$，甚至当$|K|\ge λ$时，（b）对于具有权$p$和常数$C>2$的向量的集合$K$，存在$K'\subseteq K$，使得${\rm dr}(K')\le C$和$|K'| \ge |K|^α$，其中$α= 1/ \lceil \log(p/2)/\log(C/2) \rceil$。
英文摘要：
We consider vectors from $\{0,1\}^n$. The weight of such a vector $v$ is the sum of the coordinates of $v$. The distance ratio of a set $L$ of vectors is ${\rm dr}(L):=\max \{ρ(x,y):\ x,y \in L\}/ \min \{ρ(x,y):\ x,y \in L,\ x\neq y\},$ where $ρ(x,y)$ is the Hamming distance between $x$ and $y$. We prove that (a) for every constant $λ>1$ there are no positive constants $α$ and $C$ such that every set $K$ of at least $λ^p$ vectors with weight $p$ contains a subset $K'$ with $|K'|\ge |K|^α$ and ${\rm dr}(K')\le C$, % even when $|K|\ge λ$, (b) For a set $K$ of vectors with weight $p$, and a constant $C>2$, there exists $K'\subseteq K$ such that ${\rm dr}(K')\le C$ and $|K'| \ge |K|^α$, where $α= 1/ \lceil \log(p/2)/\log(C/2) \rceil$.
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