16 problems

Fici–Saarela conjecture. Any binary word of length $n$ contains at least

Mäkelä's conjecture. There exists an infinite ternary word whose only abelian square factors are $00$, $11$, and $22$.

An abelian square is a factor formed by concatenating two anagrams of one another. An infinite word avoids abelian squares if none of its factors is an abelian square. Erdős's conj…

Effective-partition conjecture. The word $w$ contains at least

Fazekas–Mammoliti–Mercaş–Simpson conjecture. A binary word of length $n$ contains at least

Fici–Saari conjecture. Every binary word of length $n$ contains at least $\floor{n/4}$ distinct abelian square factors; equivalently,

Nonequivalent abelian-square bound. Every word of length $n$ contains $O(n^{3/2})$ nonequivalent abelian square factors.

For a circular binary word of length $n$, let $MDC(n)$ denote the minimum number of distinct abelian squares it contains. The circular minimum conjecture. The minimum number is … i…

For a binary linear word of length $n$, let $MNL(n)$ be the minimum number of non-equivalent abelian squares it contains. Fraenkel–Paterson–Simpson's conjecture. For every positive…

An abelian square is a word whose two consecutive halves are permutations of one another. Erdős's conjecture. There exists an infinite word over a four-letter alphabet containing n…

For a binary word of length $n$, let $XNL(n)$ denote the maximum number of non-equivalent abelian squares it contains. Kucherov et al.'s conjecture. … The conjecture proposes a pol…

Let a word of length $n$ contain $k$ distinct abelian square factors. Fici–Mignosi's conjecture. There exists a binary word of length $n$ containing at least $k$ distinct abelian s…

Two abelian squares are inequivalent when they have different Parikh vectors, where the Parikh vector records the number of occurrences of each letter. Let $n$ be a positive intege…

For a finite word, an abelian-square factor is a factor formed by concatenating two anagrams of one another. Let $n$ be a positive integer, and let $k$ be the number of distinct ab…

Two abelian squares are inequivalent when they have different Parikh vectors. Inequivalent-abelian-square bound conjecture. Every word of length $n$ contains $O(n\sqrt{n})$ inequiv…

For a finite word, an abelian square is a factor formed by concatenating two anagrammatic words. Binary-word maximality conjecture. If a word of length $n$ contains $k$ distinct ab…