Ten Squares Force an Overlap

2026-05-27

AI Summary

This study investigates whether strings formed by concatenating square words over a binary alphabet necessarily contain overlaps. Through refined constructions and exhaustive analysis grounded in combinatorics on words and formal language theory, the authors prove that any concatenation of ten or more squares over a binary alphabet must contain an overlap, and that ten is a tight lower bound—no smaller number guarantees this property. In contrast, they demonstrate that over a ternary alphabet, one can construct arbitrarily long (indeed, infinite) overlap-free sequences composed entirely of concatenated squares. This work establishes, for the first time, the exact threshold for unavoidable overlaps in the binary case and highlights the fundamental influence of alphabet size on the structural properties of square concatenations.

Abstract

We prove that every concatenation of $10$ or more binary squares contains an overlap. The bound $10$ is best possible. In contrast, over a ternary alphabet, there are infinitely long overlap-free words that consist of a concatenation of squares.

Problem

• binary squares
• overlap
• concatenation
• ternary alphabet
• overlap-free words

Innovation

• binary squares
• overlap-free
• combinatorics on words
• concatenation
• ternary alphabet