A unique extension of rich words

A word w is called rich if it contains vertical bar w vertical bar + 1 palindromic factors, including the empty word. We say that a rich word w can be extended in at least two ways if there are two distinct letters x, y such that wx, wy are rich. Let R denote the set of all rich words. Given w is an element of R, let K(w) denote the set of all words u such that wu is an element of R and wu can be extended in at least two ways. Let omega(w) = min{vertical bar u vertical bar vertical bar u is an element of K(w)} and let phi(n) = max{omega(w) vertical bar w is an element of R and vertical bar w vertical bar = n}, where n > 0. Vesti (2014) showed that phi(n) <= 2n. In other words, it says that for each w is an element of R there is a word u with vertical bar u vertical bar <= 2 vertical bar w vertical bar such that wu is an element of R and wu can be extended in at least two ways. We prove that phi(n) <= n and that lim sup(n ->infinity) phi(n)/n >= 2/9. The results hold for each finite alphabet having at least two letters. (C) 2021 Elsevier B.V. All rights reserved.