  In this work, we prove a $\tilde{\Omega}(\lg^{3/2} n )$ unconditional lower
bound on the maximum of the query time and update time for dynamic data
structures supporting reachability queries in $n$-node directed acyclic graphs
under edge insertions. This is the first super-logarithmic lower bound for any
natural graph problem. In proving the lower bound, we also make novel
contributions to the state-of-the-art data structure lower bound techniques
that we hope may lead to further progress in proving lower bounds.
