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The number of maximal sum-free subsets of integers Recall that the sum-free subsets of [n] described above lie in just two maximal sum-free sets This led Cameron and Erd ̋os [6] to ask whether the number of maximal sum-free subsets of [n] is “substantially smaller” than the total number of sum-free sets
Maximum subsets of (0, 1] with no solutions to x + y = kz We believe that if n is sufficiently large, to get a maximum k-sum-free subset of {1, 2, one takes the integers within the three intervals ob-tained by multiplying each real number in , n} S3(3, 1) by n (with slight modi-fication of the end-points due to integer round-off)
Maximum sum free subset - Mathematics Stack Exchange We call a set “sum free” if no two elements of the set add up to a third element of the set What is the maximum size of a sum free subset of {1, 2, …, 2n − 1} {1, 2,, 2 n 1} I am just a secondary school student I read this from a book and it is about pigeonhole principle How should I find it?
Sum-free set - Wikipedia A sum-free set is said to be maximal if it is not a proper subset of another sum-free set Let be defined by is the largest number such that any set of n nonzero integers has a sum-free subset of size k
Big Breakthrough in the exciting world of sum-free sets! a) Find f (n) that goes to infinity such that every set of n reals has a sum-free subset of size n 3 + f (n) b) Replace reals with other sets closed under addition: naturals, integers, some groups of interest
The Real Numbers - UC Davis The closed interval B = [0, 1], and the half-open interval C = (0, 1] have the same supremum and infimum as A Both sup B and inf B belong to B, while only sup C belongs to C The completeness of may be expressed in terms of the existence of suprema R Theorem 1 7 Every nonempty set of real numbers that is bounded from above has a supremum
Maximal sum-free sets in the integers Let S consist of n together with precisely one number from each pair fx; n xg for odd x < n=2 Notice distinct S lie in distinct maximal sum-free subsets of [n]
The number of maximal sum-free subsets of integers Note that both the set of odd numbers in [n] and the set fbn=2c + 1; : : : ; ng are maximal sum-free sets (A sum-free subset of [n] is maximal if it is not properly contained in another sum-free subset of [n] ) By considering all possible subsets of one of these maximal sum-free sets, we see that [n] contains at least 2dn=2e sum-free sets Cameron and Erd}os [5] conjectured that in fact [n
Maximum subsets of (0; 1] with no solutions to x + y = kz In this paper, for each real number k greater than or equal to 3 we will construct a family of k-sum-free subsets (0; 1], each of which is the union of ̄nitely many intervals (Lemma