4 problems

Fraenkel's conjecture. There is a unique, up to $\mathfrak{S}_{N}$-action, primitive circular balanced word $\mathbf{F}_{N}$ with pairwise distinct amounts of letters for…

A word is called $r$-free if it contains no factor of exponent at least $r$. Let $\operatorname{RT}(n)$ be the infimum of the real numbers $r$ for which there exist arbitrarily lon…

For each integer $k\geq 4$, let $\mathrm{RT}(k)$ be the repetition threshold, and let $\mathrm{CRT}_{\mathrm{W}}(k)$ and $\mathrm{CRT}_{\mathrm{I}}(k)$ be respectively the infima o…

For a circular word, define the strong circular repetition threshold by … Here a circular word is $\beta$-free if it has no factor of exponent at least $\beta$. Gorbunova's conject…