For a sequence S of elements from an additive abelian group G, let f(S) denote the number of subsequences of S the sum of whose terms is zero. In this paper we characterize all sequences S in G with f(S) > 2|S|−2, where |S| denotes the number of terms of S. MSC Classification : Primary 11B50, Secondary 11B65, 11B75.

A graph G is called a sum graph if there exists a labelling of the vertices of G by distinct positive integers such that the vertices labelled u and v are adjacent if and only if there exists a vertex labelled u + v. If G is not a sum graph, adding a nite number of isolated vertices to it will always yield a sum graph, and the sum number (G) of G is the smallest number of isolated vertices that...

We count the ordered sum-free triplets of subsets in group $\mathbb{Z}/p\mathbb{Z}$, i.e., $(A,B,C)$ sets $A,B,C \subset \mathbb{Z}/p\mathbb{Z}$ for which equation $a+b=c$ has no solution with $a\in A$, $b \in B$ and $c C$. Our main theorem improves on a recent result by Semchankau, Shabanov, Shkredov using different simpler method. proof relates previous results number independent regular grap...

In this paper we give a survey of recent results on sumsets of subsets of a field with polynomial restrictions. Let F be a field and let F× be the multiplicative group F \ {0}. The additive order of the (multiplicative) identity of F is either infinite or a prime, we call it the characteristic of F . Let A and B be finite subsets of the field F . Set A+B = {a+ b: a ∈ A and b ∈ B} and AuB = {a+ ...