Related papers: k-fold Sidon sets

Let A={a_s(mod n_s)}_{s=0}^k be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results concerning system A. In particular, we show that if every integer lies in…

In this paper we show that for any $k\geq2$, there exist two universal constants $C_k,D_k>0$, such that for any finite subset $A$ of positive real numbers with $|AA|\leq M|A|$, $|kA|\geq \frac{C_k}{M^{D_k}}\cdot|A|^{\log_42k}.$

We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $\Omega(n^{1-1/s})$ must contain a nontrivial solution to every fixed translation-invariant linear equation in at least five variables. This extends…

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every…

We show that if A is a subset of {1,...,N} containing no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{3/4-o(1)} N).

It was shown by V. Bergelson that any set B with positive upper multiplicative density contains nicely intertwined arithmetic and geometric progressions: For each positive integer k there exist integers a,b,d such that $ {b(a+id)^j:i,j…

Consider families of $k$-subsets (or blocks) on a ground set of size $v$. Recall that if all $t$-subsets occur with the same frequency $\lambda$, one obtains a $t$-design with index $\lambda$. On the other hand, if all $t$-subsets occur…

We derive lower and upper bounds on possible growth rates of certain sets of positive integers $A_k=\{1= a_1 < a_2 < ... < a_{k}\}$ such that all integers $n\in \{0, 1, 2, ..., ka_{k}\}$ can be represented as a sum of no more than $k$…

Let $TT_k$ denote the transitive tournament on $k$ vertices. Let $TT(h,k)$ denote the graph obtained from $TT_k$ by replacing each vertex with an independent set of size $h \geq 1$. The following result is proved: Let $c_2=1/2$, $c_3=5/6$…

We show that while the number of coprime compositions of a positive integer $n$ into $k$ parts can be expressed as a $\mathbb{Q}$-linear combinations of the Jordan totient functions, this is never possible for the coprime partitions of $n$…

We study a conjecture by Deaconescu on the solubility of finite groups with claims that if more than half of the elements in a finite group has the same order $k$, then the group is soluble. We show that the original conjecture fails by…

Let $f(n,k)$ be the largest number of positive integers not exceeding $n$ from which one cannot select $k+1$ pairwise coprime integers, and let $E(n,k)$ be the set of positive integers which do not exceed $n$ and can be divided by at least…

A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k-1)-dimensional skeleton and \binom{n-1}{k} facets such that H_k(X;Q)=0. Here we introduce the following family of simplicial complexes. Let n,k be integers…

Let $A$ be a finite set of integers. We show that if $k$ is a prime power or a product of two distinct primes then $$|A+k\cdot A|\geq(k+1)|A|-\lceil k(k+2)/4\rceil$$ provided $|A|\geq (k-1)^{2}k!$, where $A+k\cdot A=\{a+kb:\ a,b\in A\}$. We…

For a fixed integer $k \ge 0$, consider representations of positive integers as sums of binomial coefficients of the form $\binom{n}{k}$. While exact minimal bounds for the number of required summands are known only in a few low-dimensional…

A classical theorem of Baranyai states that, given integers $2\leq k < n$ such that $k$ divides $n$, one can find a family of ${n-1\choose k-1}$ partitions of $[n]$ into $k$-element subsets such that every subset appears in exactly one…

Let k>2 be a fixed integer exponent and let \theta > 9/10. We show that a positive integer N can be represented as a non-trivial sum or difference of 3 k-th powers, using integers of size at most B, in O(B^{\theta}N^{1/10}) ways, providing…

Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether…

Let N be a large enough natural number, A and B be subsets of {N+1, ... , 2N}. In this paper, we prove that there exists integers a, b with a belongs to A, b belongs to B such that ab=P_k^2 + O(P_k^{1-c}), where 0<c<1/2 and P_k denotes an…

For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets. The size of the…