Extremal problems and the combinatorics of sumsets

For finite subsets A_1,...,A_n of a field, their sumset is given by {a_1+...+a_n: a_1 in A_1,...,a_n in A_n}. In this paper we study various restricted sumsets of A_1,...,A_n with restrictions of the following forms: a_i-a_j not in S_{ij}, or alpha_ia_i not=alpha_ja_j, or a_i+b_i not=a_j+b_j (mod m_{ij}). Furthermore, we gain an insight into relations among recent results on this area obtained in quite different ways.

Let $h\geq 2$ and $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a finite set of integers. It is well-known that $\left|hA\right|=hk-h+1$ if and only if $A$ is a $k$-term arithmetic progression. In this paper, we give some nontrivial inverse results of the sets $A$ with some extrema the cardinalities of $hA$.

Let $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n\}$$ and $$C(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_i\in A_i\ (1\le i\le n),\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n,\ \text{and}\ a_n\not=a_1\}$$ recently introduced by the second author, when $G...

We consider two problems regarding some divisibility properties of the subset sums of a set $A\subseteq \{1, 2, \ldots ,n\}$. At the beginning, we study the cardinality of $A$ which has the following property: For every $d\le n$ there is a non empty set $A_d\subseteq A$ 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 ...

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.

Let $h\geq 2$ be a positive integer. For any subset $\mathcal{A}\subset \mathbb{Z}_n$, let $h^{\wedge}\mathcal{A}$ be the set of the elements of $\mathbb{Z}_n$ which are sums of $h$ distinct elements of $\mathcal{A}$. In this paper, we obtain some new results on $4^{\wedge}\mathcal{A}$ and $5^{\wedge}\mathcal{A}$. For example, we show that if $|\mathcal{A}|\geq 0.4045n$ and $n$ is odd, then $4^{\wedge}\mathcal{A}=\mathbb{Z}_{n}$; Under some conditions, if $n$ is even and $|...

Given a set A in Z/NZ we may form a Cayley sum graph G_A on vertex set Z/NZ by joining i to j if and only if i + j is in A. We investigate the extent to which performing this construction with a random set A simulates the generation of a random graph, proving that the clique number of G_A is a.s. O(log N). This shows that Cayley sum graphs can furnish good examples of Ramsey graphs. To prove this result we must study the specific structure of set addition on Z/NZ. Indeed, we ...

In the paper we find new inequalities involving the intersections $A\cap (A-x)$ of shifts of some subset $A$ from an abelian group. We apply the inequalities to obtain new upper bounds for the additive energy of multiplicative subgroups and convex sets and also a series another results on the connection of the additive energy and so--called higher moments of convolutions. Besides we prove new theorems on multiplicative subgroups concerning lower bounds for its doubling consta...

We say the sets of nonnegative integers A and B are additive complements if their sum contains all sufficiently large integers. In this paper we prove a conjecture of Chen and Fang about additive complement of a finite set.

In this paper, we consider certain finite sums related to the "largest odd divisor", and we obtain, using simple ideas and recurrence relations, sharp upper and lower bounds for these sums.