Ten Squares Force an Overlap
arXiv:2605.285707.5
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Settles a combinatorial problem about unavoidable patterns in binary strings, providing a tight bound.
The paper proves that any concatenation of 10 or more binary squares must contain an overlap, and shows that the bound 10 is optimal. Over a ternary alphabet, infinitely long overlap-free words composed of squares exist.
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.