On solution-free sets of integers II
نویسندگان
چکیده
منابع مشابه
Enumerating solution-free sets in the integers
Given a linear equation L, a set A ⊆ [n] is L-free if A does not contain any ‘non-trivial’ solutions to L. In this paper we consider the following three general questions: (i) What is the size of the largest L-free subset of [n]? (ii) How many L-free subsets of [n] are there? (iii) How many maximal L-free subsets of [n] are there? We completely resolve (i) in the case when L is the equation px ...
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Given a linear equation L, a set A ⊆ [n] is L-free if A does not contain any ‘non-trivial’ solutions to L. We determine the precise size of the largest L-free subset of [n] for several general classes of linear equations L of the form px+ qy = rz for fixed p, q, r ∈ N where p ≥ q ≥ r. Further, for all such linear equations L, we give an upper bound on the number of maximal L-free subsets of [n]...
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A set S of integers is said to be sum-free if a, b e 5 implies a + b 6 S. In this paper, we investigate two new problems on sum-free sets: (1) Let f(k) denote the largest positive integer for which there exists a partition of (1, 2,... ,f(k)) into k sum-free sets, and let h(k) denote the largest positive integer for which there exists a partition of {1, 2, . . . , h(k)) into k sets which are su...
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Given a linear equation L, a set A of integers is L-free if A does not contain any ‘nontrivial’ solutions to L. This notion incorporates many central topics in combinatorial number theory such as sum-free and progression-free sets. In this paper we initiate the study of (parameterised) complexity questions involving L-free sets of integers. The main questions we consider involve deciding whethe...
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where a,, and b, equal 0 or 1, the set A (resp. B) appearing as the set of those n such that a,, = 1 (resp. b, = 1). It seems a rather difficult problem to describe explicitly the structure of all such direct factors, although the theorem demonstrated in [5] and our present Theorem 1 shed some light on the situation by proving the existence of their asymptotic densities. (The corresponding addi...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2017
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa8522-6-2017