Related papers: Largest Sidon subsets in weak Sidon sets

Given a size-$k$ subset $S$ of a group $G$, how large can the product set $S^n$ be? We study this question, at several layers of refinement, for the infinite dihedral group. First, we give an explicit formula for the maximum size of $S^n$…

A set of natural numbers is primitive if no element of the set divides another. Erd\H{o}s conjectured that if S is any primitive set, then \sum_{n\in S} 1/(n log n) \le \sum_{n\in \P} 1/(p log p), where \P denotes the set of primes. In this…

Let $F_n$ be a free group of rank $n$, with free generating set $X$. A subset $D$ of $F_n$ is a \emph{Distinct Difference Configuration} if the differences $g^{-1}h$ are distinct, where $g$ and $h$ range over all (ordered) pairs of distinct…

Let $G$ be an abelian group. A set $A \subset G$ is a \emph{$B_k^+$-set} if whenever $a_1 + \dots + a_k = b_1 + \dots + b_k$ with $a_i, b_j \in A$ there is an $i$ and a $j$ such that $a_i = b_j$. If $A$ is a $B_k$-set then it is also a…

More than 50 years ago, Erd\H os asked the following question: what is the maximum size of a family $\mathcal F$ of $k$-element subsets of an $n$-element set if it has no $s+1$ pairwise disjoint sets? This question attracted a lot of…

We study s(k,n), the maximum size of A+A where A is a k-subset of [n]. A few known functions from additive number theory can be expressed via s(k,n). For example, our estimates of s(k,n) imply new bounds on the maximum size of quasi-Sidon…

A family $\mathcal{F}$ of subsets of $[n]=\{1,2,\ldots,n\}$ shatters a set $A \subseteq [n]$ if for every $A' \subseteq A$ there is an $F \in \mathcal{F}$ such that $F \cap A=A'$. We develop a framework to analyze $f(n,k,d)$, the maximum…

Let $G$ be a finite abelian group of order $n$, and for each $a\in G$ and integer $1\le h\le n$ let $\mathcal{F}_a(h)$ denote the family of all $h$-element subsets of $G$ whose sum is $a$. A problem posed by Katona and Makar-Limanov is to…

We say that the weak width of an infinite subgroup $H$ of $G$ in $G$ is $n$ if there exists a collection of $n$ strongly essentially distinct conjugates $\{ H, g_1^{-1} H g_1,\cdots, g_{n-1}^{-1} H g_{n-1} \}$ of $H$ in $G$ such that the…

We prove the following variant of the Erd\H{o}s distinct subset sums problem. Given $t \ge 0$ and sufficiently large $n$, every $n$-element set $A$ whose subset sums are distinct modulo $N=2^n+t$ satisfies $$\max A \ge…

Let $hA$ denote the $h$-fold sumset of a subset $A$ of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations $\sigma_1, \ldots, \sigma_H \in \mathfrak{S}_n$, there exist finite subsets $A_1,…

A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erd\H{o}s raised the problem of determining the maximum possible size of a union of $r$ different intersecting families of…

Let $Y$ be a compact metric space, $G$ be a group acting by transformations on $Y$. For any infinite subset $A\subset Y$, we study the density of $gA$ for $g\in G$ and quantitative density of the set $\displaystyle{\bigcup_{g\in G_n}gA}$ by…

We resolve a $1000 Erd\H{o}s prize problem, complete with formal verification generated by a large language model. In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erd\H{o}s repeatedly posed one of his "favourite"…

For a family $\mathcal{F}$ of sets and a disjoint pair $A,B$ we let $\mathcal{F}(A,\overline{B})=\{F\in \mathcal{F}: A\subseteq F, ~B\cap F=\emptyset\}$. The \textbf{$(p,q)$-d\"omd\"od\"om} of a family $\mathcal{F}\subseteq 2^{[n]}$ is…

For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, given a finite almost simple group $G$ and any maximal subgroup $H$ of $G$, we determine a precise upper bound for $d(H)$. In…

Let $G$ be the alternating group of degree $n$. Let $\omega(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $\sigma(G)$ be the minimal size of a family of proper…

In answer to a question raised recently by Bourgain and Lewko, we show, with their paper's terminology, that any uniformly bounded $\psi_2 (C)$-orthonormal system ($\psi_2 (C)$ is a variant of subGaussian)is 2-fold tensor Sidon. This…

Erdos, Herzog and Piranian asked whether, for $n$ points in the plane with fixed diameter (maximum distance between points), an arrangement of a regular $n$-gon maximizes their product of all pairs of distances. Recently, it was discovered…

We prove that if $A$ is an infinite multiplicative Sidon set, then $\liminf\limits_{n\to \infty}\frac{|A(n)|-\pi (n)}{\frac{n^{3/4}}{(\log n)^3}}<\infty$ and construct an infinite multiplicative Sidon set satisfying $\liminf\limits_{n\to…