Related papers: Sum of elements in finite Sidon sets II

Let $h \geq 2$ and let ${ \mathcal A} = (A_1,\ldots, A_h)$ be an $h$-tuple of sets of integers. For nonzero integers $c_1,\ldots, c_h$, consider the linear form $\varphi = c_1 x_1 + c_2x_2 + \cdots + c_h x_h$. The \emph{representation…

A family $\{A_{0},\ldots,A_{d}\}$ of $k$-element subsets of $[n]=\{1,2,\ldots,n\}$ is called a simplex-cluster if $A_{0}\cap\cdots\cap A_{d}=\varnothing$, $|A_{0}\cup\cdots\cup A_{d}|\le2k$, and the intersection of any $d$ of the sets in…

For a finite group $G$, let $m_I(G)$ denote the largest possible cardinality of a minimal invariable generating set of $G$. We prove an upper and a lower bound for $m_I(S_n)$, which show in particular that $m_I(S_n)$ is asymptotic to $n/2$…

Let $G$ be a finite additive abelian group with exponent $n$ and $S=g_{1}\cdots g_{t}$ be a sequence of elements in $G$. For any element $g$ of $G$ and $A\subseteq\{1,2,\ldots,n-1\}$, let $N_{A,g}(S)$ denote the number of subsequences…

Let $D$ be a subset of a finite commutative ring $R$ with identity. Let $f(x)\in R[x]$ be a polynomial of positive degree $d$. For integer $0\leq k \leq |D|$, we study the number $N_f(D,k,b)$ of $k$-subsets $S\subseteq D$ such that…

It is a classical fact that every $n$-element set of positive reals has at least $\binom{n+1}{2}+1$ distinct subset sums, with equality exactly for homogeneous arithmetic progressions (when $n\geq 4$). We establish stability versions of…

Let $\Z_m$ be the group of residue classes modulo $m$. Let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups of the group $\Z_m \times \Z_n$ and the number of its cyclic subgroups, respectively, where $m$ and $n$ are arbitrary…

One form of the inclusion-exclusion principle asserts that if A and B are functions of finite sets then A(S) is the sum of B(T) over all subsets T of S if and only if B(S) is the sum of (-1)^|S-T| A(T) over all subsets T of S. If we replace…

Studies on partition of $I_n$ = $\{1, 2, . . . , n\}$ into subsets $S_1, S_2, . . . , S_x$ so far considered with prescribed sum of the elements in each subset. In this paper, we study constant sum partitions $\{S_1,S_2,...,S_x\}$ of $I_n$…

Erd\H{o}s and Graham proposed to determine the number of subsets $S \subseteq \left\{1,2,\dots,n\right\}$ with $\sum_{s \in S} 1/s = 1$ and asked, among other things, whether that number could be as large as $2^{n - o(n)}$. We show that the…

A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = A+A = \{a + b \ | \ a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. We prove that the number of sumsets in $\mathbb{F}_2^n$ is asymptotically…

Let (G, *) be a semigroup, D subset of G, and n >= 2 be an integer. We say that (D, *) is an n-closed subset of G if a_1* ... *a_n in D for every a_1, ..., a_n in D. Hence every closed set is a 2-closed set. The concept of n-closed sets…

Applying the enumeration of sparse set partitions, we show that the number of set systems H such that the emptyset is not in H, the total cardinality of edges in H is n, and the vertex set of H is {1, 2, ..., m}, equals (1/log(2)+o(1))^nb_n…

The set A = {a_n} of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be represented as the sum of h elements of A. If a_n ~ alpha n^h for some real number alpha > 0, then alpha is called an…

Let $S = \{p_1,\dots,p_r\}$ be a finite set of distinct primes, let $\Psi_S(X)$ be the number of $S$-smooth integers not exceeding $X$, and let $F_S(X)$ be the maximum size of a subset of $M(S) \cap [1,X]$ containing no set $\{n,p_1…

Nathanson introduced the range of cardinalities of $h$-fold sumsets $R(h,k) := \{|hA|:A \subset \mathbb{Z} \text{ and }|A| = k\}.$ Following a remark of Erd\H{o}s and Szemer\'edi that determined the form of $R(h,k)$ when $h=2$, Nathanson…

The classical Ulam sequence is defined recursively as follows: $a_1=1$, $a_2=2$, and $a_n$, for $n > 2$, is the smallest integer not already in the sequence that can be written uniquely as the sum of two distinct earlier terms. This…

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…

For a set $A$ of positive integers with $\gcd(A)=1$, let $\langle A \rangle$ denote the set of all finite linear combinations of elements of $A$ over the non-negative integers. The it is well known that only finitely many positive integers…

An $S_k$-set in a group $\Gamma$ is a set $A\subseteq\Gamma$ such that $\alpha_1\cdots\alpha_k=\beta_1\cdots\beta_k$ with $\alpha_i,\beta_i\in A$ implies $(\alpha_1,\ldots,\alpha_k)=(\beta_1,\ldots,\beta_k)$. An $S_k'$-set is a set such…