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 $m\le n$ words of length $n$ then, for every $k\ge n+m$, it contains at most $⌈\left(m+1\right)/2⌉⌊\left(m+1\right)/2⌋$ 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/

[1] S. Ferenczi, Rank and symbolic complexity. Ergodic Theory Dyn. Syst. 16 (1996) 663-682. | Zbl 0858.68051

[2] S. Ferenczi, Complexity of sequences and dynamical systems. Discrete Math. 206 (1999) 145-154. | Zbl 0936.37008

[3] N.J. Fine and H.S. Wilf, Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16 (1965) 109-114. | Zbl 0131.30203

[4] A. Heinis, The $P\left(n\right)/n$-function for bi-infinite words. Theoret. Comput. Sci. 273 (2002) 35-46. | Zbl 1027.68654

[5] T. Kamae and L. Zamboni, Sequence entropy and the maximal pattern complexity of infinite words. Ergodic Theory Dynam. Syst. 22 (2002) 1191-1199. | Zbl 1014.37004

[6] M. Morse and A.G. Hedlund, Symbolic dynamics. Amer. J. Math 60 (1938) 815-866. | JFM 64.0798.04

[7] R. Tijdeman, Periodicity and almost periodicity | MR 2223402 | Zbl 1103.68103