منابع مشابه
Integer Sets with Distinct Subset - Sums
In Section 1 we introduce the problem of finding minimal-height sets of n natural numbers with distinct subset-sums (SSD), and in Section 2 review the well-known Conway-Guy sequence u, conjectured to yield a minimal SSD set for every n. We go on (Section 3) to prove that u certainly cannot be improved upon by any "greedy" sequence, to verify numerically (Section 4) that it does yield SSD sets f...
متن کاملA Construction for Sets of Integers with Distinct Subset Sums
A set S of positive integers has distinct subset sums if there are 2|S| distinct elements of the set {∑ x∈X x : X ⊂ S } . Let f(n) = min{maxS : |S| = n and S has distinct subset sums}. Erdős conjectured f(n) ≥ c2n for some constant c. We give a construction that yields f(n) < 0.22002 · 2n for n sufficiently large. This now stands as the best known upper bound on f(n).
متن کاملBalanced Subset Sums in Dense Sets of Integers
Let 1 ≤ a1 < a2 < · · · < an ≤ 2n − 2 denote integers. Assuming that n is large enough, we prove that there exist ε1, . . . , εn ∈ {−1,+1} such that |ε1 + · · ·+εn| ≤ 1 and |ε1a1+ · · ·+εnan| ≤ 1. This result is sharp, and in turn it confirms a conjecture of Lev. We also prove that when n is even, every integer in a large interval centered at (a1 + a2 + · · · + an)/2 can be represented as the s...
متن کاملSets of Unit Vectors with Small Subset Sums
We say that a family {xi | i ∈ [m]} of vectors in a Banach space X satisfies the k-collapsing condition if ‖∑i∈I xi‖ ≤ 1 for all k-element subsets I ⊆ {1,2, . . . ,m}. Let C (k,d) denote the maximum cardinality of a k-collapsing family of unit vectors in a d-dimensional Banach space, where the maximum is taken over all spaces of dimension d. Similarly, let C B(k,d) denote the maximum cardinalit...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1993
ISSN: 0012-365X
DOI: 10.1016/0012-365x(93)90378-7