  Recent work has established that, for every positive integer $k$, every
$n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges that is
resilient to $f$ edge or vertex faults. For vertex faults, this bound is tight.
However, the case of edge faults is not as well understood: the best known
lower bound for general $k$ is $\Omega(f^{\frac12 - \frac{1}{2k}} n^{1+1/k}
+fn)$. Our main result is to nearly close this gap with an improved upper
bound, thus separating the cases of edge and vertex faults. For odd $k$, our
new upper bound is $O_k(f^{\frac12 - \frac{1}{2k}} n^{1+1/k} + fn)$, which is
tight up to hidden $poly(k)$ factors. For even $k$, our new upper bound is
$O_k(f^{1/2} n^{1+1/k} +fn)$, which leaves a gap of $poly(k) f^{1/(2k)}$. Our
proof is an analysis of the fault-tolerant greedy algorithm, which requires
exponential time, but we also show that there is a polynomial-time algorithm
which creates edge fault tolerant spanners that are larger only by factors of
$k$.
