A temporal graph is a graph in which every edge carries a non-empty set of time labels, and it is temporally connected if for every two vertices $u$ and $v$, there exists a $u$-$v$-path with non-decreasing time labels. A spanner is a subset of its edges preserving temporal connectivity. Unlike static graphs, temporally connected graphs need not admit sparse spanners; nonetheless, minimizing spanner size is a central and widely studied problem. A particularly intriguing question is whether temporal cliques admit spanners of linear size. Despite considerable effort over the past years, the best known upper bound remained $O(n \log n)$. We finally resolve this question, proving that every temporal clique on $n$ vertices admits a spanner of size $7n$. Moreover, such a spanner can be computed in polynomial time.

翻译：时序图是一种每条边都带有一个非空时间标签集合的图，若对于任意两个顶点 $u$ 和 $v$，存在一条时间标签非递减的 $u$-$v$ 路径，则称其是时序连通的。生成子是保留时序连通性的边子集。与静态图不同，时序连通图未必允许稀疏的生成子；尽管如此，最小化生成子的规模仍是一个核心且被广泛研究的问题。一个尤为引人注目的问题是：时序团是否允许线性规模的生成子？尽管过去几年付出了大量努力，已知的最佳上界仍为 $O(n \log n)$。我们最终解决了这一问题，证明了任意 $n$ 个顶点的时序团均可容纳规模为 $7n$ 的生成子。此外，这样的生成子可在多项式时间内计算得到。