3 problems

Samsonov–Shur's conjecture. The exact values are

Petrova–Shur's conjecture. The Abelian repetition thresholds satisfy

Let $\mathcal{A}_n$ be an alphabet of size $n$, and let a crucial word be a word avoiding abelian $k$-th powers such that appending any letter from $\mathcal{A}_n$ creates an abeli…