Hereditary properties of words

Balogh, József; Bollobás, Béla

doi:10.1051/ita:2005003

Balogh, József ; Bollobás, Béla

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 49-65.

Résumé

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 $m \leq n$ words of length $n$ then, for every $k \geq n + m$ , it contains at most $⌈ (m + 1) / 2 ⌉ ⌊ (m + 1) / 2 ⌋$ words of length $k$ .

EuDML MR Zbl

DOI : 10.1051/ita:2005003

Classification : 05C
Mots clés : graph properties, monotone, hereditary, speed, size

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     title = {Hereditary properties of words},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {49--65},
     publisher = {EDP-Sciences},
     volume = {39},
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Balogh, József; Bollobás, Béla. Hereditary properties of words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 49-65. doi : 10.1051/ita:2005003. http://www.numdam.org/articles/10.1051/ita:2005003/

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