A new approach to nonrepetitive sequences

A sequence is nonrepetitive if it does not contain two adjacent identical blocks. The remarkable construction of Thue asserts that 3 symbols are enough to build an arbitrarily long nonrepetitive sequence. It is still not settled whether the following extension holds: for every sequence of 3-element sets L1, . . . , Ln there exists a nonrepetitive sequence s1, . . . , sn with si ∈ Li. Applying the probabilistic method one can prove that this is true for sufficiently large sets Li. We present an elementary proof that sets of size 4 suffice (confirming the best known bound [14]). The argument is a simple counting with Catalan numbers involved. Our approach is inspired by a new algorithmic proof of the Lov´asz Local Lemma due to Moser and Tardos [19] and its interpretations by Fortnow [10] and Tao [22]. The presented method has further applications to nonrepetitive games and nonrepetitive colorings of graphs.

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Jaroslaw Grytczuk, Jakub Kozik, Piotr Micek
• Keywords: Adjacent Identical Blocks, CORR 2011, Education, Long Nonrepetitive Sequence, Nonrepetitive Sequence