Related papers: On the maximum values of the additive representati…

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…

Consider a finite field $\mathbb{F}_q$ and positive integers $d,m,r$ with $1\leq r\leq \binom{m+d}{d}$. Let $S_d(m)$ be the $\mathbb{F}_q$ vector space of all homogeneous polynomials of degree $d$ in $X_0,\dots,X_m$. Let $e_r(d,m)$ be the…

Let $S^r(n)$ be the $r$-graph on $n$ vertices with parts $A$ and $B$, where the edges consist of all $r$-tuples with $1$ vertex in $A$ and $r-1$ vertices in $B$, and the sizes of $A$ and $B$ are chosen to maximise the number of edges. Let…

Let $\mathrm{ex}(n,s)$ denote the maximum number of edges in a triangle-free graph on $n$ vertices which contains no independent sets larger than $s$. The behaviour of $\mathrm{ex}(n,s)$ was first studied by Andr\'asfai, who conjectured…

If $s$ is a positive integer and $A$ is a set of positive integers, we say that $B$ is an $s$-divisor of $A$ if $\sum_{b\in B} b\mid s\sum_{a\in A} a$. We study the maximal number of $k$-subsets of an $n$-element set that can be…

Let $F$ be a graph. We say that a hypergraph $H$ contains an induced Berge $F$ if the vertices of $F$ can be embedded to $H$ (e.g., $V(F)\subseteq V(H)$) and there exists an injective mapping $f$ from the edges of $F$ to the hyperedges of…

Let $\mathcal{R}_k(n)$ be the number of representations of an integer $n$ as the sum of a prime and a $k$-th power. Define E_k(X) := |\{n \le X, n \in I_k, n\text{not a sum of a prime and a $k$-th power}\}|. Hardy and Littlewood conjectured…

We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n) \ll n^\alpha$ for…

We show that for every rational number $r \in (1,2)$ of the form $2 - a/b$, where $a, b \in \mathbb{N}^+$ satisfy $\lfloor b/a \rfloor^3 \le a \le b / (\lfloor b/a \rfloor +1) + 1$, there exists a graph $F_r$ such that the Tur\'an number…

Let $\mathcal{F}$ be a family of $r$-graphs. The Tur\'an number $ex_r(n;\mathcal{F})$ is defined to be the maximum number of edges in an $r$-graph of order $n$ that is $\mathcal{F}$-free. The famous Erd\H{o}s Matching Conjecture shows that…

Let $s(m,n)$ denote the classical \DED sum, where $n$ is a positive integer and $m\in\{0,1,\ldots, n-1\}$, $(m,n)=1$. For a given positive integer $k$, we describe a set of at most $k^2$ numbers $m$ for which $s(m,n)$ may be $\ge s(k,n)$,…

Let k_1,...,k_d be positive integers, and D be a subset of [k_1]x...x[k_d], whose complement can be decomposed into disjoint sets of the form {x_1}x...x{x_{s-1}}x[k_s]x{x_{s+1}}x...x{x_d}. We conjecture that the number of elements of D can…

For two graphs $G$ and $H$, the Tur\'{a}n number $ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Tur\'{a}n numbers $ex(K_{m,n},kK_2)$ for all $k\geq 1$ [7]. In this…

A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or…

The maximum correlation of functions of a pair of random variables is an important measure of stochastic dependence. It is known that this maximum nonlinear correlation is identical to the absolute value of the Pearson correlation for a…

The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $(r-2)$ new vertices. In this paper we determine…

It is well known that Erd\H{o}s Matching Conjecture concerns the maximum number of hyperedges in an $r$-uniform hypergraph with bounded matching number. As a generalization, it is natural to ask for the maximum number of copies of…

Consider nondeterministic finite automata recognizing base-k positional notation of numbers. Assume that numbers are read starting from their least significant digits. It is proved that if two sets of numbers S and T are represented by…

We prove that $d_k(n)=d_k(n+B)$ infinitely often for any positive integers $k$ and $B$, where $d_k(n)$ denotes the number of divisors of $n$ coprime to $k$.

The classical problem of maximizing the Shannon entropy of a sum of independent random variables supported on a finite alphabet is considered and settled in the ternary case. Namely, the following theorem is established: if…