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Article 13.2.7 On Highly Repetitive and Power Free Words
"... Answering a question of Richomme, Currie and Rampersad proved that 7/3 is the infimum of the real numbers α> 2 such that there exists an infinite binary word that avoids αpowers but is highly 2repetitive, i.e., contains arbitrarily large squares beginning at every position. In this paper, we pr ..."
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Answering a question of Richomme, Currie and Rampersad proved that 7/3 is the infimum of the real numbers α> 2 such that there exists an infinite binary word that avoids αpowers but is highly 2repetitive, i.e., contains arbitrarily large squares beginning at every position. In this paper, we
A PROOF OF DEJEAN’S CONJECTURE
, 905
"... Abstract. We prove Dejean’s conjecture. Specifically, we show that Dejean’s conjecture holds for the last remaining open values of n, namely 15 ≤ n ≤ 26. 1. ..."
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Abstract. We prove Dejean’s conjecture. Specifically, we show that Dejean’s conjecture holds for the last remaining open values of n, namely 15 ≤ n ≤ 26. 1.
Avoiding Approximate Squares
 DLT 2007, FINLANDE
, 2007
"... As is wellknown, Axel Thue constructed an infinite word over a 3letter alphabet that contains no squares, that is, no nonempty subwords of the form xx. In this paper we consider a variation on this problem, where we try to avoid approximate squares, that is, subwords of the form xx′ where x = ..."
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As is wellknown, Axel Thue constructed an infinite word over a 3letter alphabet that contains no squares, that is, no nonempty subwords of the form xx. In this paper we consider a variation on this problem, where we try to avoid approximate squares, that is, subwords of the form xx′ where x = x′ and x and x′ are “nearly” identical.
A note on avoidable words in squarefree ternary words
, 2008
"... We completely characterize the words that can be avoided in infinite squarefree ternary words. 1 ..."
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We completely characterize the words that can be avoided in infinite squarefree ternary words. 1
Dejean’s conjecture holds for n ≥ 30
"... We extend Carpi’s results by showing that Dejean’s conjecture holds for n ≥ 30. The following definitions are from sections 8 and 9 of [1]: Fix n ≥ 30. Let m = ⌊(n − 3)/6⌋. Let Am = {1, 2,..., m}. Let ker ψ = {v ∈ A ∗ m∀a ∈ Am, 4 divides va}. (In fact, this is not Carpi’s definition of ker ψ, but ..."
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We extend Carpi’s results by showing that Dejean’s conjecture holds for n ≥ 30. The following definitions are from sections 8 and 9 of [1]: Fix n ≥ 30. Let m = ⌊(n − 3)/6⌋. Let Am = {1, 2,..., m}. Let ker ψ = {v ∈ A ∗ m∀a ∈ Am, 4 divides va}. (In fact, this is not Carpi’s definition of ker ψ, but rather the assertion of his Lemma 9.1.) A word v ∈ A + m is a ψkernel repetition if it has period q and a prefix v ′ of length q such that v ′ ∈ ker ψ, (n−1)(v+1) ≥ nq − 3. It will be convenient to have the following new definition: If v has period q and its prefix v ′ of length q is in ker ψ, we say that q is a kernel period of v. As Carpi states at the beginning of section 9 of [1]: By the results of the previous sections, at least in the case n ≥ 30, in order to construct an infinite word on n letters avoiding
A note on nonrepetitive colourings of planar graphs, arXiv:math/0307365 v 1
, 2003
"... Alon et al. introduced the concept of nonrepetitive colourings of graphs. Here we address some questions regarding nonrepetitive colourings of planar graphs. Specifically, we show that the faces of any outerplanar map can be nonrepetitively coloured using at most five colours. We also give some l ..."
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Alon et al. introduced the concept of nonrepetitive colourings of graphs. Here we address some questions regarding nonrepetitive colourings of planar graphs. Specifically, we show that the faces of any outerplanar map can be nonrepetitively coloured using at most five colours. We also give some lower bounds for the number of colours required to nonrepetitively colour the vertices of both outerplanar and planar graphs. 1
Dejean’s conjecture holds for n ≥ 27
, 2009
"... We show that Dejean’s conjecture holds for n ≥ 27. Repetitions in words have been studied since the beginning of the previous century [13, 14]. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 5, 6, 7, 9]. For rational 1 < r ≤ 2, a fractional rpower is a non ..."
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We show that Dejean’s conjecture holds for n ≥ 27. Repetitions in words have been studied since the beginning of the previous century [13, 14]. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 5, 6, 7, 9]. For rational 1 < r ≤ 2, a fractional rpower is a nonempty word w = xx ′ such that x ′ is the prefix of x of length (r − 1)x. For example, 010 is a 3/2power. A basic problem is that of identifying the repetitive threshold for each alphabet size n> 1: What is the infimum of r such that an infinite sequence on n letters exists, not containing any factor of exponent greater than r? The infimum is called the repetitive threshold of an nletter alphabet, denoted by RT(n). Dejean’s conjecture [5] is that ⎨ 7/4, n = 3 RT(n) = 7/5, n = 4 n/(n − 1) n ̸ = 3, 4 Thue, Dejean and Pansiot, respectively [14, 5, 12] established the values RT(2), RT(3), RT(4). MoulinOllagnier [11] verified Dejean’s conjecture for 5 ≤ n ≤ 11, and MohammadNoori and Currie [10] proved the conjecture for 12 ≤ n ≤ 14.
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