Related papers: The counting version of a problem of Erd\H{o}s

A family of subsets $\mathcal{F}\subseteq {[n]\choose k}$ is called intersecting if any two of its members share a common element. Consider an intersecting family, a direct problem is to determine its maximal size and the inverse problem is…

The maximum size, $La(n,P)$, of a family of subsets of $[n]=\{1,2,...,n\}$ without containing a copy of $P$ as a subposet, has been intensively studied. Let $P$ be a graded poset. We say that a family $\mathcal{F}$ of subsets of…

In this paper we study the Product Partition Problem (PPP), i.e. we are given a set of $n$ natural numbers represented on $m$ bits each and we are asked if a subset exists such that the product of the numbers in the subset equals the…

In this paper, using the compression method, we recover the lower bound for the Erd\H{o}s unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, in $\mathbb{R}^k$ for all $k\geq 2$, we…

Propositions 1.1 -- 1.3 stated below contribute to results and certain problems considered in a paper by Erdos and Szekeres, on the behavior of products $\prod^n_1 (1-z^{a_j}), 1\leq a_1\leq \cdots\leq a_n$ integers. In the discussion,…

We study the following generalization of the Tur\'an problem in sparse random graphs. Given graphs $T$ and $H$, let $\mathrm{ex}\big(G(n,p), T, H\big)$ be the random variable that counts the largest number of copies of $T$ in a subgraph of…

For a set $A$ of $k$ elements from an additive abelian group $G$ and a positive integer $r \leq k$, we consider the set of elements of $G$ that can be written as a sum of $h$ elements of $A$ with at least $r$ distinct elements. We denote…

Let $\mathcal{A}\subset\mathbb{R}_{\geqslant1}$ be a countable set such that $\limsup_{x\to\infty}\frac{1}{\log x}\sum_{\alpha\in\mathcal{A}\cap[1,x]}\frac{1}{\alpha}>0$. We prove that, for every $\varepsilon>0$, there exist infinitely many…

Let $La(n,P)$ be the maximum size of a family of subsets of $[n]=\{1,2,...,n\}$ not containing $P$ as a (weak) subposet. The diamond poset, denoted $B_{2}$, is defined on four elements $x,y,z,w$ with the relations $x<y,z$ and $y,z<w$.…

This paper deals with syzygies of Segre embeddings. Let d >=3 and n_1, ..., n_d nonzero natural numbers. We prove that O(1,...., 1) on the product of P^{n_1}, ...,P^{n_d} satisfies Property N_p if and only if p <= 3.

The classical Erd\H os-Ko-Rado theorem states that if $k\le\floor{n/2}$ then the largest family of pairwise intersecting $k$-subsets of $[n]=\{0,1,...,n\}$ is of size ${{n-1}\choose{k-1}}$. A family of $k$ subsets satisfying this pairwise…

We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. Using flag algebras we obtain new numerical results on the Erd\H{o}s problem of finding…

Motivated by intuitions from projective algebraic geometry, we provide a novel construction of subsets of the $d$-dimensional grid $[n]^d$ of size $n - o(n)$ with no $d + 2$ points on a sphere or a hyperplane. For $d = 2$, this improves the…

Let $\mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of $\mathbb{Z}_n$, namely, the maximum size of a collection of permutations such that the sum of any two distinct…

Pach showed that every $d+1$ sets of points $Q_1,\dotsc,Q_{d+1} \subset \mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. We show, by means of an example, that a…

The matching number of a family of subsets of an $n$-element set is the maximum number of pairwise disjoint sets. The families with matching number $1$ are called intersecting. The famous Erd\H os-Ko-Rado theorem determines the size of the…

Two families $\mathcal{A}$ and $\mathcal{B}$, of $k$-subsets of an $n$-set, are {\em cross $t$-intersecting} if for every choice of subsets $A \in \mathcal{A}$ and $B \in \mathcal{B}$ we have $|A \cap B| \geq t$. We address the following…

Let $p$ be a prime number, and $h$ a positive integer such that $\gcd(p,h)=1$. We prove, without invoking Dirichlet's theorem, that the arithmetic progression $p\left(\mathbf{N}\cup \{0\}\right)+h$ contains infinitely many prime numbers.…

For an integer $n \geq 1$, the Erd\H{o}s-Rogers function $f_{s}(n)$ is the maximum integer $m$ such that every $n$-vertex $K_{s+1}$-free graph has a $K_s$-free subgraph with $m$ vertices. It is known that for all $s \geq 3$, $f_{s}(n) =…

In this paper we study combinatorial aspects of permutations of $\{1,\ldots,n\}$ and related topics. In particular, we prove that there is a unique permutation $\pi$ of $\{1,\ldots,n\}$ such that all the numbers $k+\pi(k)$ ($k=1,\ldots,n$)…