The number of maximal sum-free subsets of integers
نویسندگان
چکیده
Cameron and Erdős [6] raised the question of how many maximal sum-free sets there are in {1, . . . , n}, giving a lower bound of 2bn/4c. In this paper we prove that there are in fact at most 2(1/4+o(1))n maximal sum-free sets in {1, . . . , n}. Our proof makes use of container and removal lemmas of Green [8, 9] as well as a result of Deshouillers, Freiman, Sós and Temkin [7] on the structure of sum-free sets.
منابع مشابه
Sharp bound on the number of maximal sum-free subsets of integers
Cameron and Erdős [6] 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. Here, 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))2 n/4 maximal sum-free...
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Our main topic is the number of subsets of 1; n] which are maximal with respect to some condition such as being sum-free, having no number dividing another, etc. We also investigate some related questions. In our earlier paper 8], we considered conditions on sets of positive integers (sum-freeness, Sidon sequences, etc.), and attempted to estimate the number of subsets of 1; n] satisfying each ...
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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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