  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.
