We show that for every $k$-dimensional linear code $\mathcal{C} \subseteq \mathbb{F}_2^n$ there exists a set $S\subseteq [n]$ of size at most $n/2+O(\sqrt{nk})$ such that the projection of $\mathcal{C}$ onto $S$ has distance at least $\frac12\mathrm{dist}(\mathcal{C})$. As a consequence we show that any connected graph $G$ with $m$ edges and $n$ vertices has at least $2^{m-(n-1)}$ many $1/2$-thin subgraphs.

翻译：我们证明对于每个$k$维线性码$\mathcal{C} \subseteq \mathbb{F}_2^n$，存在一个大小至多为$n/2+O(\sqrt{nk})$的集合$S\subseteq [n]$，使得$\mathcal{C}$在$S$上的投影距离至少为$\frac12\mathrm{dist}(\mathcal{C})$。由此我们进一步证明，任何具有$m$条边和$n$个顶点的连通图$G$至少包含$2^{m-(n-1)}$个$1/2$-薄子图。