Simon's congruence, denoted ∼ n , relates words having the same subwords of length up to n. We show that, over a k-letter alphabet, the number of words modulo ∼ n is in 2 Θ(n k−1 log n) .

Post Embedding Problems are a family of decision problems based on the interaction of a rational relation with the subword embedding ordering, and are used in the literature to prove non multiply-recursive complexity lower bounds. We refine the construction of Chambart and Schnoebelen (LICS 2008) and prove parametric lower bounds depending on the size of the alphabet.