Every binary pattern of length six is avoidable on the two-letter alphabet

U. Schmidt [9, 10] showed that every pattern on two letters of length at least 13 is avoidable an a two-letter alphabet (i.e. 2-avoidable). We prove that this bound can be improved to 6. Since there are patterns of length 5 being not 2-avoidable, this bound is optimal.

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