  We prove that there exists an absolute constant $\delta>0$ such any binary
code $C\subset\{0,1\}^N$ tolerating $(1/2-\delta)N$ adversarial deletions must
satisfy $|C|\le 2^{\text{poly}\log N}$ and thus have rate asymptotically
approaching 0. This is the first constant fraction improvement over the trivial
bound that codes tolerating $N/2$ adversarial deletions must have rate going to
0 asymptotically. Equivalently, we show that there exists absolute constants
$A$ and $\delta>0$ such that any set $C\subset\{0,1\}^N$ of $2^{\log^A N}$
binary strings must contain two strings $c$ and $c'$ whose longest common
subsequence has length at least $(1/2+\delta)N$. As an immediate corollary, we
show that $q$-ary codes tolerating a fraction $1-(1+2\delta)/q$ of adversarial
deletions must also have rate approaching 0.
  Our techniques include string regularity arguments and a structural lemma
that classifies binary strings by their oscillation patterns. Leveraging these
tools, we find in any large code two strings with similar oscillation patterns,
which is exploited to find a long common subsequence.
