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 disjoint identical (scattered) subwords of length m. Let f(n,Σ)=min{f(S):S is of length n, over alphabet Σ}. Here, it is shown that2f(n,{0,1})=n−o(n) using the regularity lemma for words. In other words, 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.