The zero-rate threshold for adversarial bit-deletions is less than 1/2

We prove that there exists an absolute constant $delta>0$ such any binary code $Csubset{0,1}^N$ 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 $Csubset{0,1}^N$ 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 $1-(1+2delta)/q$ 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.