Strict bounds for pattern avoidance

Cassaigne conjectured in 1994 that any pattern with m   distinct variables of length at least 3(2m−1)3(2m−1) is avoidable over a binary alphabet, and any pattern with m   distinct variables of length at least 2m2m is avoidable over a ternary alphabet. Building upon the work of Rampersad and the power series techniques of Bell and Goh, we obtain both of these suggested strict bounds. Similar bounds are also obtained for pattern avoidance in partial words, sequences where some characters are unknown.