On the number of maximal sum-free sets
نویسندگان
چکیده
منابع مشابه
Bounds on the Number of Maximal Sum-Free Sets
We show that the number of maximal sum-free subsets of {1, 2, . . . , n} is at most 2. We also show that 2 is an upper bound on the number of maximal product-free subsets of any group of order n.
متن کاملOn the Number of Maximal Sum-free Sets
It is shown that the set {1, 2, . . . , n} contains at most 2n/2−2n maximal sum-free subsets, provided n is large enough. A set A ⊆ [n] = {1, 2, . . . , n} is sum-free if for any two elements a, b ∈ A we have a + b / ∈ A. A sum-free set A ⊆ [n] is maximal if it is not contained in any other sum-free subset of [n]. Let s(n) and smax(n) denote the number of sum-free and maximal sum-free subsets o...
متن کاملSmall Maximal Sum-Free Sets
Let G be a group and S a non-empty subset of G. If ab / ∈ S for any a, b ∈ S, then S is called sum-free. We show that if S is maximal by inclusion and no proper subset generates 〈S〉 then |S| ≤ 2. We determine all groups with a maximal (by inclusion) sum-free set of size at most 2 and all of size 3 where there exists a ∈ S such that a / ∈ 〈S \ {a}〉.
متن کاملOn the maximal density of sum-free sets
Theorem 1 is due to Folkman [4], who also asked whether its assertion remains true if ε > 0 is replaced by a function which tends to 0 as n→∞. Theorem 2 below states that this is indeed the case and, furthermore, for every set A dense enough, one can take b = 0. It should be mentioned that recently a similar result has been independently proved by Hegyvári [5], who showed that the assertion of ...
متن کاملA sharp bound on the number of maximal sum-free sets
Cameron and Erdős [7] asked whether the number of maximal sum-free sets in {1, . . . , n} is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of 2bn/4c for the number of maximal sum-free sets. We prove the following: For each 1 ≤ i ≤ 4, there is a constant Ci such that, given any n ≡ i mod 4, {1, . . . , n} contains (Ci + o(1))2n/4 maximal sum-free sets. ...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2000
ISSN: 0002-9939,1088-6826
DOI: 10.1090/s0002-9939-00-05815-9