A large family of words must contain two words that are similar. We
investigate several problems where the measure of similarity is the length of a
common subsequence.
We construct a family of n^{1/3} permutations on n letters, such that LCS of
any two of them is only cn^{1/3}, improving a construction of Beame, Blais, and
Huynh-Ngoc. We relate the problem of constructing many permutations with small
LCS to the twin word problem of Axenovich, Person and Puzynina. In particular,
we show that every word of length n over a k-letter alphabet contains two
disjoint equal subsequences of length cnk^{-2/3}.
Many problems are left open.Comment: 18+epsilon page