Application of Entropy Compression in Pattern Avoidance

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $f$ of $w$ such that $f= h(p)$ where $h: \Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We give a positive answer to Problem 3.3.2 in Lothaire's book "Algebraic combinatorics on words'", that is, every pattern with $k$ variables of length at least $2^k$ (resp. $3\times2^{k-1}$) is 3-avoidable (resp. 2-avoidable). This conjecture was first stated by Cassaigne in his thesis in 1994. This improves previous bounds due to Bell and Goh, and Rampersad.