Résumé
We characterize the formulas that are avoided by every$\alpha$ -free word for some$\alpha>1$ . We study the avoidability index of formulas whose fragments are of the form$XYX$ . The largest avoidability index of an avoidable palindrome pattern is known to be at least$4$and at most$16$ . We make progress toward the conjecture that every avoidable palindrome pattern is$4$ -avoidable.