  Let $G=(V,E)$ be an undirected graph on $n$ vertices and $\lambda:E\to
2^{\mathbb{N}}$ a mapping that assigns to every edge a non-empty set of integer
labels (times). Such a graph is {\em temporally connected} if a path exists
with non-decreasing times from every vertex to every other vertex. In a seminal
paper, Kempe, Kleinberg, and Kumar \cite{KKK02} asked whether, given such a
temporal graph, a {\em sparse} subset of edges always exists whose labels
suffice to preserve temporal connectivity -- a {\em temporal spanner}. Axiotis
and Fotakis \cite{AF16} answered negatively by exhibiting a family of
$\Theta(n^2)$-dense temporal graphs which admit no temporal spanner of density
$o(n^2)$. In this paper, we give the first positive answer as to the existence
of $o(n^2)$-sparse spanners in a dense class of temporal graphs, by showing
(constructively) that if $G$ is a complete graph, then one can always find a
temporal spanner of density $O(n \log n)$.
