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A famous conjecture of Erd\H{o}s asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem…

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…

It is well-known that any sequence of at least N integers contains a subsequence whose sum is 0 (mod N). However, there can be very few subsequences with this property (e.g. if the initial sequence is just N 1's, then there is only one…

Let $k$ be an integer which is the difference between prime numbers infinitely often. It is known that there are infinitely many such $k$ and, in this paper, we give a new unconditional proof that these $k$ have positive density and improve…

For $x>0$ let $\pi(x)$ denote the number of primes not exceeding $x$. For integers $a$ and $m>0$, we determine when there is an integer $n>1$ with $\pi(n)=(n+a)/m$. In particular, we show that for any integers $m>2$ and $a\le\lceil…

Let $\mathcal{A}$ be a union-closed family of sets with universe $\bigcup_{A \in \mathcal{A}}A = [n] = \{1,\cdots,n\}$ and length $\ell$. We prove that $|\mathcal{A}| \leq \sum_{i=0}^{\ell} \binom{n}{i}$, with equality if and only if…

For $m\geq 1$, let $0<b_0<b_1<...<b_m$ and $\ e_0,e_1,...,e_m>0$ be fixed positive integers. Assume there exists a prime $p$ and an integer $t>0$ such that $p^t\mid b_0$, but $p^t\nmid b_{i}\ {\rm for}\ 1\leq i\leq m$. Then, we prove that…

Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every strict subpattern $P'$…

The $abc$ conjecture states that there are only finitely many triples of coprime positive integers $(a,b,c)$ such that $a+b=c$ and $\operatorname{rad}(abc) < c^{1-\epsilon}$ for any $\epsilon > 0$. Using the optimized methods in a recent…

We say that a family ${x_i|i\in[m]}$ of vectors in a Banach space $X$ satisfies the $k$-collapsing condition if $|\sum_{i\in I}x_i|\leq 1$ for all $k$-element subsets $I\subseteq{1,2,...,m}$. Let $C(k,d)$ denote the maximum cardinality of a…

We present and analyze an algorithm to enumerate all integers $n\le x$ that can be written as the sum of consecutive $k$th powers of primes, for $k>1$. We show that the number of such integers $n$ is asymptotically bounded by a constant…

Paul Erd\H{o}s and L\'{a}szl\'{o} Lov\'{a}sz proved in a landmark article that, for any positive integer $k$, up to isomorphism there are only finitely many maximal intersecting families of $k-$sets (maximal $k-$cliques). So they posed the…

The Kahane--Salem--Zygmund inequality for multilinear forms in $\ell_{\infty}$ spaces claims that, for all positive integers $m,n_{1},...,n_{m}$, there exists an $m$-linear form $A\colon\ell_{\infty}^{n_{1}}\times\cdots\times…

We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{\sigma(1)} \cdots x_{\sigma(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $\sigma$…

It is conjectured since long that for any convex body $K \subset \mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at least $2n$ normals from different points on the boundary of $K$. The conjecture is known to be…

Given a rational $a=p/q$ and $N$ nonnegative $d$-dimensional real vectors $u_1$, ..., $u_N$, we show that it is always possible to choose $(d-1)+\lceil (pN-d+1)/q\rceil$ of them such that their sum is (componentwise) at least…

We show the existence of a constant $c > 0$ such that, for all positive integers $n$, there exist integers $1 \leq a_1 < \ldots < a_k \leq n$ such that there are at least $cn^2$ distinct integers of the form $\sum_{i=u}^{v}a_i$ with $1 \leq…

Let BS(m,n) denote the set of base sequences (A;B;C;D), with A and B of length m and C and D of length n. The base sequence conjecture (BSC) asserts that BS(n+1,n) exist (i.e., are non-empty) for all n. This is known to be true for n <= 36…

It has been known since Erdos that the sum of $1/(n\log n)$ over numbers $n$ with exactly $k$ prime factors (with repetition) is bounded as $k$ varies. We prove that as $k$ tends to infinity, this sum tends to 1. Banks and Martin have…

We prove that the ratio of the Newman sum over numbers multiple of a fixed integer which is not multiple of 3 and the Newman sum over numbers multiple of a fixed integer divisible by 3 is o(1) when the upper limit of summing tends to…