COMBINATORICA Bolyai Society – Springer-Verlag 0209–9683/103/$6.00 c 2003 J´ anos Bolyai Mathematical Society Combinatorica 23 (4) (2003) 571–584 THE FIXING BLOCK METHOD IN COMBINATORICS ON WORDS JAMES D. CURRIE, CAMERON W. PIERCE Received February 1, 2000 We give an overview of the method of fixing blocks introduced by Shelton. We apply the method to words which are nonrepetitive up to mod k. 1. Introduction A word is repetitive if it contains two consecutive identical blocks. For ex- ample, barbarian = barbarian is repetitive, while civilized is non-repetitive. A word containing k consecutive identical blocks is said to contain a k power. Thus a repetitive word contains a 2 power. Dejean [4] introduced the study of words containing fractional k powers. For example, the word civilized contains ivi =(iv) 3/2 ,a3/2 power. Thue showed [9] that there are infinite words over the three letter al- phabet {a,b,c} which are non-repetitive. Infinite non-repetitive words (se- quences) have been used to build counter-examples in algebra [7], ordered sets [10], symbolic dynamics [6] and other areas. In constructing general- izations of the 1-dimensional non-repetitive sequences to higher dimensions, Currie and Simpson [5] introduced sequences which are non-repetitive up to mod r. A sequence {s n } ∞ n=1 is non-repetitive up to mod r if each of its mod k subsequences {s nk+j } ∞ n=1 is non-repetitive, 1 ≤ k ≤ r,0 ≤ j ≤ k - 1. A non-empty set L of infinite words is perfect if for any u ∈ L and any n there is a word v ∈ L, v = u such that u and v have a common prefix of Mathematics Subject Classification (2000): 05-XX, 68Q45