Related papers: A note on iterated sumsets races

The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their…

The n-th crossing number of a graph G, denoted cr_n(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a>b>0, there exists a graph G for which cr_0(G) = a, cr_1(G) = b, and…

The bipartite Ramsey number $B(n_1,n_2,\ldots,n_t)$ is the least positive integer $b$ such that, any coloring of the edges of $K_{b,b}$ with $t$ colors will result in a monochromatic copy of $K_{n_i,n_i}$ in the $i-$th color, for some $i$,…

We present a variety of new results on finite sets A of integers for which the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a…

Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order $n$ equals $A(n):={{1!4!7! ... (3n-2)!} \over {n!(n+1)! ... (2n-1)!}}$. Mills, Robbins, and Rumsey also made the stronger…

This paper aims at finding conditions on a Hamburger or Stieltjes moment sequence, under which the change of at most a finite number of its entries produces another sequence of the same type. It turns out that a moment sequence allows all…

For any set $A$ of natural numbers with positive upper Banach density, we show the existence of an infinite set $B$ and sequences $(t_k)_{k\in \mathbb{N}}, (s_k)_{k\in \mathbb{N}}$ of natural numbers such that $\left\{ \sum_{n \in F}n : F…

Let $\gamma_n$ be the permutation on $n$ symbols defined by $\gamma_n = (1\ 2\...\ n)$. We are interested in an enumerative problem on colored permutations, that is permutations $\beta$ of $n$ in which the numbers from 1 to $n$ are colored…

Given a base $b$, a "digit map" is a map $f: \mathbb{Z}^{\ge 0} \to \mathbb{Z}^{\ge 0}$ of the form $f(\sum_{i=0}^n a_ib^i) = \sum_{i=0}^n f_*(a_i)$, $0 \le a_i \le b-1$ for each $i$, where $f_* : \{0,1,\dots, b-1\} \to \mathbb{Z}^{\ge 0}$…

A hereditary property of combinatorial structures is a collection of structures (e.g. graphs, posets) which is closed under isomorphism, closed under taking induced substructures (e.g. induced subgraphs), and contains arbitrarily large…

A set $\mathcal{A}$ is said to be an additive $h$-basis if each element in $\{0,1,\ldots,hn\}$ can be written as an $h$-sum of elements of $\mathcal{A}$ in {\it at least} one way. We seek multiple representations as $h$-sums, and, in this…

A conjecture posed by S. Hayajneh and F. Kittaneh claims that given $A,B$ positive matrices, $0\le t\le 1$, and any unitarily invariant norm it holds $|||A^tB^{1-t}+B^tA^{1-t}|||\le|||A^tB^{1-t}+A^{1-t}B^t|||$. Recently, R. Bhatia proved…

We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\{1, 2, \dots, n\}$ of size at least $C n^{1/k}$, then, for some $d \leq k-1$, the set of subset sums of $A$ contains a…

A Ramsey-like theorem is a statement of the form ``For every 2-coloring of $[\mathbb{N}]^2$, there exists an infinite set~$H \subseteq \mathbb{N}$ such that $[H]^2$ avoids some pattern''. We prove that none of these statements are…

A family of permutations $A \subset S_n$ is said to be \emph{$t$-set-intersecting} if for any two permutations $\sigma, \pi \in A$, there exists a $t$-set $x$ whose image is the same under both permutations, i.e. $\sigma(x)=\pi(x)$. We…

By definition, the intersection of finitely many open sets of any topological space is open. Nachbin observed that, more generally, the intersection of compactly many open sets is open. Moreover, Nachbin applied this to obtain elegant…

Motivated by questions asked by Erdos, we prove that any set $A\subset{\mathbb N}$ with positive upper density contains, for any $k\in{\mathbb N}$, a sumset $B_1+\cdots+B_k$, where $B_1,\dots,B_k\subset{\mathbb N}$ are infinite. Our proof…

We provide an internal characterization of those finite algebras (i.e., algebraic structures) $\mathbf A$ such that the number of homomorphisms from any finite algebra $\mathbf X$ to $\mathbf A$ is bounded from above by a polynomial in the…

We show that if $A=\{a_1,a_2,..., a_k\}$ is a monotone increasing set of numbers, and the differences of the consecutive elements are all distinct, then $|A+B|\geq c|A|^{1/2}|B|$ for any finite set of numbers $B$. The bound is tight up to…

Motivated by an extremal problem on graph-codes that links coding theory and graph theory, Alon recently proposed a question aiming to find the smallest number $t$ such that there is an edge coloring of $K_{n}$ by $t$ colors with no copy of…