A regularity lemma and twins in words

Abstract: For a word $S$, let $f(S)$ be the largest integer $m$ such that there are two disjoints identical (scattered) subwords of length $m$. Let $f(n, \Sigma) = \min {f(S): S \text{is of length} n, \text{over alphabet} \Sigma }$. Here, it is shown that [2f(n, {0,1}) = n-o(n)] using the regularity lemma for words. I.e., any binary word of length $n$ can be split into two identical subwords (referred to as twins) and, perhaps, a remaining subword of length $o(n)$. A similar result is proven for $k$ identical subwords of a word over an alphabet with at most $k$ letters.