4 problems

For each $k\geqslant 2$, let $\mathcal{G}_k$ be the graph whose vertices are square-free words over a $k$-letter alphabet, with edges arising from the square-reduction relation. Fi…

For an alphabet with $k$ letters, let $\deg^+(W)$ be the number of words obtainable from $W$ by a single square reduction, and let $\deg_k^+(n)$ be the maximum value of $\deg^+(W)$…

For an alphabet with $k$ letters, let $d(W)$ be the minimum number of single square reductions needed to turn a word $W$ into a square-free word, and let $d_k(n)$ be the maximum va…

Let a ternary word be a word over a three-letter alphabet, and let $\mathcal{R}(W)$ denote the set of distinct reducts of a word $W$. Missing-values conjecture. There exist infinit…