Some Variations on a Theme of Irina Mel'nichuk Concerning the Avoidability of Patterns in Strings of Symbols.

The set of all doubled patterns on n or fewer letters can be avoided on an alphabet with k letters, where k is the least even integer strictly greater than n+1, with the exception of n = 4. The set of all doubled patterns on 4 or fewer letters can be avoided on the 8-letter alphabet. The set of all avoidable patterns on n or fewer letters can be avoided on an alphabet with 2(n + 2) letters.