On a conjecture of Alon

We consider two problems regarding some divisibility properties of the subset sums of a set A ⊆ {1, 2,. .. , n}. At the beginning, we study the cardinality of A which has the following property: For every d ≤ n there is a non empty set A d ⊆ A such that the sum of the elements of A d is a multiple of d. Next, we turn our attention to another problem: If all subset sums of A form a multiple free-sequence, what can we say about the structure of A? We give some asymptotics for the first problem and improve some already existing results for the second one.

Combinatorics, Probability and Computing, 1999

Let A be a subset of an abelian group G. The subset sum of A is the set [sum ](A) = {[sum ]x∈T[mid ]T⊂A}. We prove the following result. Let S be a generating subset of an abelian group G such that 0∉S and 14[les ][mid ]S[mid ]. Then one of the following conditions holds.(i) [mid ][sum ](S)[mid ][ges ]min([mid ]G[mid ] −3, 3[mid ]S[mid ]−3).(ii) There is an x∈S such that S[setmn ]{x} generates a proper subgroup of order less than (3[mid ]S[mid ]−3)/2.As a consequence, we obtain the following open case of an old conjecture of Diderrich. Let q be a composite odd number and let G be an abelian group of order 3q. Let S be a subset of G with cardinality q+1. Then every element of G is the sum of some subset of S.

Discrete Mathematics, 1990

We prove that for positive integers n, m and k, the set (1, 2,

arXiv (Cornell University), 2021

For a given positive integer k, the Sidon-Ramsey number SR(k) is defined as the minimum value of n such that, in every partition of the set [1, n] into k parts, there exists a part that contains two distinct pairs of numbers with the same sum. In other words, there is a part that is not a Sidon set. In this paper, we investigate the asymptotic behavior of this parameter and two generalizations of it. The first generalization involves replacing pairs of numbers with h-tuples, such that in every partition of [1, n] into k parts, there exists a part that contains two distinct h-tuples with the same sum. Alternatively, there is a part that is not a B h set. The second generalization considers the scenario where the interval [1, n] is substituted with a nonnecessarily symmetric d-dimensional box of the form d i=1 [1, n i ]. For the general case of h ≥ 3 and non-symmetric boxes, before applying our method to obtain the Ramsey-type result, we needed to establish an upper bound for the corresponding density parameter.

Integers, 2006

We define a sum as a set {x, y, z} of distinct natural numbers such that x + y = z, and let N m = {1, 2,. .. , m}. We introduce a new sequence h(n) defined as the smallest s such that there is no partition of N s into n sum-free parts. We determine h(n) for n = 3, 4 after easily noting that h(1) = 3 and h(2) = 9. We find that h(3) = 24 and h(4) = 67 using a computer program.