In combinatorics on words, a word w over an alphabet Σ is said to avoid a pattern p over an alphabet Δ if there is no factor f of w such that f=h(p) where h:Δ∗→Σ∗ 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 2k (resp. 3×2k−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.