Related papers: A New Upper Bound for Cancellative Pairs

A cube tiling of $\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t\colon t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t)=\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$…

Let $V \subset \mathbb{R}$ be a finite set with $|V| = n $ and suppose we are given each pairwise distance independently with probability $p$. We show that if $p = (1+\epsilon)/n$, for some fixed $\epsilon >0$, then we can reconstruct a…

We study the number of $s$-element subsets $J$ of a given abelian group $G$, such that $|J+J|\leq K|J|$. Proving a conjecture of Alon, Balogh, Morris and Samotij, and improving a result of Green and Morris, who proved the conjecture for $K$…

A recent conjecture of Caputo, Carlen, Lieb, and Loss, and, independently, of the author, states that the maximum of the permanent of a matrix whose rows are unit vectors in l_p is attained either for the identity matrix I or for a constant…

We show that the number of all maximal $\alpha$-gapped repeats and palindromes of a word of length $n$ is at most $3(\pi^2/6 + 5/2) \alpha n$ and $7 (\pi^2 / 6 + 1/2) \alpha n - 5 n - 1$, respectively.

A circular word, or a necklace, is an equivalence class under conjugation of a word. A fundamental question concerning regularities in standard words is bounding the number of distinct squares in a word of length $n$. The famous conjecture…

A family of disjoint pairs of finite sets $\mathcal{P}=\{(A_i,B_i)\mid i\in[m]\}$ is called a Bollob\'as system if $A_i\cap B_j\neq\emptyset$ for every $i\neq j$, and a skew Bollob\'as system if $A_i\cap B_j\neq\emptyset$ for every $i<j$.…

Write ${\cal I}(G)$ for the set of independent sets of a graph $G$ and $i(G)$ for $|{\cal I}(G)|$. It has been conjectured (by Alon and Kahn) that for an $N$-vertex, $d$-regular graph $G$, $$ i(G) \leq \left(2^{d+1}-1\right)^{N/2d}. $$ If…

For nonnegative integers $n_2, n_3$ and $d$, let $N(n_2,n_3,d)$ denote the maximum cardinality of a code of length $n_2+n_3$, with $n_2$ binary coordinates and $n_3$ ternary coordinates (in this order) and with minimum distance at least…

A Sidon set is a set of integers containing no nontrivial solutions to the equation $a+b=c+d$. We improve on the lower bound on the diameter of a Sidon set with $k$ elements: if $k$ is sufficiently large and ${\cal A}$ is a Sidon set with…

In this paper we describe all pairs of binary vectors $({\bf u}, {\bf v})$ such that the set of vectors obtained by $t$ deletions in ${\bf v}$ is a subset of the set of vectors obtained by $t$ deletions in ${\bf u}$ for $t=1,2$. Such pairs…

Sumsets are central objects in additive combinatorics. In 2007, Granville asked whether one can efficiently recognize whether a given set $S$ is a sumset, i.e. whether there is a set $A$ such that $A+A=S$. Granville suggested an algorithm…

For $k\geq3$, a collection of $k$ sets is said to form a \emph{weak $\Delta$-system} if the intersection of any two sets from the collection has the same size. Erd\H{o}s and Szemer\'{e}di asked about the size of the largest family…

Given an irreducible fraction $\frac{c}{d} \in [0,1]$, a pair $(\mathcal{A},\mathcal{B})$ is called a $\frac{c}{d}$-cross-intersecting pair of $2^{[n]}$ if $\mathcal{A}, \mathcal{B}$ are two families of subsets of $[n]$ such that for every…

Let $X$ and $X'$ be affine algebraic varieties over a field $\mathbb{k}$. The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism $X\times\mathbb{A}^n\cong X'\times\mathbb{A}^n$ implies $X\cong X'$. In…

We prove two conjectures in this paper. The first conjecture is by Lund, Pham and Thu: Given a Borel set $A\subset \mathbb{R}^n$ such that $\dim A\in (k,k+1]$ for some $k\in\{1,\dots,n-1\}$. For $0<s<k$, we have \[ \text{dim}(\{y\in…

A set of integers $S \subset \mathbb{N}$ is an $\alpha$-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on $\alpha$, more specifically if $| (x+w) - (y+z) | \geq \max \{…

We obtain an upper bound for the number of pairs $ (a,b) \in {A\times B} $ such that $ a+b $ is a prime number, where $ A, B \subseteq \{1,...,N \}$ with $|A||B| \, \gg \frac{N^2}{(\log {N})^2}$, $\, N \geq 1$ an integer. This improves on a…

Let ${\cal H}(A,B)$ denote the set of homomorphisms from the poset $A$ to the poset $B$. In previous studies, the author has started to analyze what it is in the structure of finite posets $R$ and $S$ that results in $# {\cal H}(P,R) \leq #…

More than twenty-five years ago, Manickam, Miklos, and Singhi conjectured that for positive integers $n,k$ with $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum is…