Hereditary properties of words
Let P be a hereditary property of words, i.e., an infinite class of finite words such that every subword (block) of a word belonging to P is also in P. Extending the classical Morse-Hedlund theorem, we show that either P contains at least n + 1 words of length n for every n or, for some N, it contains at most N words of length n for every n. More importantly, we prove the following quantitative extension of this result: if P has m ≤ n words of length n then, for every k ≥ n + m, it contains at most [(m + 1)/2] [(m + 1)/2] words of length k. © EDP Sciences 2005.
RAIRO - Theoretical Informatics and Applications
Balogh, J., & Bollobás, B. (2005). Hereditary properties of words. RAIRO - Theoretical Informatics and Applications, 39 (1), 49-65. https://doi.org/10.1051/ita:2005003