Hereditary properties of words
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 39 (2005) no. 1, p. 49-65

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 mn words of length n then, for every kn+m, it contains at most (m+1)/2(m+1)/2 words of length k.

DOI : https://doi.org/10.1051/ita:2005003
Classification:  05C
Keywords: graph properties, monotone, hereditary, speed, size
@article{ITA_2005__39_1_49_0,
     author = {Balogh, J\'ozsef and Bollob\'as, B\'ela},
     title = {Hereditary properties of words},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {1},
     year = {2005},
     pages = {49-65},
     doi = {10.1051/ita:2005003},
     zbl = {1132.68048},
     mrnumber = {2132578},
     language = {en},
     url = {http://www.numdam.org/item/ITA_2005__39_1_49_0}
}
Balogh, József; Bollobás, Béla. Hereditary properties of words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 39 (2005) no. 1, pp. 49-65. doi : 10.1051/ita:2005003. http://www.numdam.org/item/ITA_2005__39_1_49_0/

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