Online version of the theorem of Thue

TL;DR

This paper explores an online game version of Thue's theorem, demonstrating that Alice can construct arbitrarily long nonrepetitive sequences with at least 12 symbols, using graph coloring techniques, while shorter sequences are impossible with fewer symbols.

Contribution

It introduces an online variant of Thue's theorem and proves that Alice can build arbitrarily long nonrepetitive sequences with at least 12 symbols, using nonrepetitive graph colorings.

Findings

Alice can construct arbitrarily long nonrepetitive sequences with ≥12 symbols.

Over 4 symbols, Alice cannot play for too long.

The minimum number of symbols needed for an infinite sequence remains unknown.

Abstract

A sequence S is nonrepetitive if no two adjacent blocks of S are the same. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3 symbols. We consider the online variant of this result in which a nonrepetitive sequence is constructed during a play between two players: Bob is choosing a position in a sequence and Alice is inserting a symbol on that position taken from a fixed set A. The goal of Bob is to force Alice to create a repetition, and if he succeeds, then the game stops. The goal of Alice is naturally to avoid that and thereby to construct a nonrepetitive sequence of any given length. We prove that Alice has a strategy to play arbitrarily long provided the size of the set A is at least 12. This is the online version of the Theorem of Thue. The proof is based on nonrepetitive colorings of outerplanar graphs. On the other hand, one can prove that even over 4 symbols Alice has no chance to play for too long. The minimum size of the set of symbols needed for the online version of Thue's theorem remains unknown.