Related papers: A numerical note on upper bounds for b 2 [g] sets

A skew Bollob\'{a}s system $\mathcal{P}=\{(A_i,B_i):1\leq i\leq m\}$ is a collection of pairs of disjoint subsets of $[n]$ such that $A_i\cap B_j\ne\emptyset$ for any $1\leq i<j\leq m$. Denote by $S_1(a, b)$ or $S_2(a, b)$ the maximum size…

Let $p>7$ be a prime, let $G=\Z/p\Z$, and let $S_1=\prod_{i=1}^p g_i$ and $S_2=\prod_{i=1}^p h_i$ be two sequences with terms from $G$. Suppose that the maximum multiplicity of a term from either $S_1$ or $S_2$ is at most $\frac{2p+1}{5}$.…

Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this paper is to investigate the lower bound for…

It is shown that the maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=4$, and constant dimension $k=4$ is at most $272$. In Finite Geometry terms, the maximum number of solids in…

We prove that every infinite, discrete abelian group admits a pair of $I_0$ sets whose union is not $I_0$. In particular, this implies that every such group contains a Sidon set that is not $I_{0}$.

Let $A=(a_1, a_2, ..., a_n)$ be relative prime positive integers with $a_i\geq 2$. The Frobenius number $g(A)$ is the greatest integer not belonging to the set $\big\{ \sum_{i=1}^na_ix_i\ |x_i\in \mathbb{N}\big\}$. The general Frobenius…

We prove that if $d \ge 2$ is an integer, $G$ is a finite abelian group, $Z_0$ is a subset of $G$ not contained in any strict coset in $G$, and $E_1,\dots,E_d$ are dense subsets of $G^n$ such that the sumset $E_1+\dots+E_d$ avoids $Z_0^n$…

Let $E^n$ denote the (real) $n$-dimensional Euclidean space. It is not known whether an equilateral set in the $\ell_1$ sum of $E^a$ and $E^b$, denoted here as $E^a \oplus_1 E^b$, has maximum size at least $\dim(E^a \oplus_1 E^b) + 1 = a +…

How many odd numbers are there? How many even numbers? From Galileo to Cantor, the suggestion was that there are the same number of odd, even and natural numbers, because all three sets can be mapped in one-one fashion to each other. This…

Let $(R,m,k)$ be a Golod ring. We show a recurrent formula for high syzygies of $k$ interms of previous ones. In the case of embedding dimension at most $2$, we provided complete descriptions of all indecomposable summands of all syzygies…

A 3-simplex is a collection of four sets A_1,...,A_4 with empty intersection such that any three of them have nonempty intersection. We show that the maximum size of a set system on n elements without a 3-simplex is $2^{n-1} +…

We consider the set $$\mathcal{A} = \left\{10\cdot a + 11\cdot b \ | \gcd(a,b)=1, a\geq 1, b\geq 2a+1 \right\}.$$ We will prove that $\mathcal{A}$ is unbounded and that there exists a natural number $M\notin \mathcal{A}$ for which…

This paper is concerned with the problem of finding two sets of integers, $\{a_1, a_2, \ldots$, $a_m\}$ and $\{b_1, b_2, \ldots, b_n\}$, such that all the $mn$ sums $a_i+b_j, i=1, \ldots, m, j=1, \ldots, n$, are perfect squares. A method is…

We improve the previously best known lower and upper bounds on the number n_g of numerical semigroups of genus g. Starting from a known recursive description of the tree T of numerical semigroups, we analyze some of its properties and use…

We prove that if $A=\{a_1,\dots ,a_{|A|}\}\subset \{1,2,\dots ,n\}$ is a Sidon set so that $|A|=n^{1/2}-L^\prime$, then $$a_m = m\cdot n^{1/2} + \mathcal O\left( n^{7/8}\right) + \mathcal O\left(L^{1/2}\cdot n^{3/4}\right)$$ where…

Let $A \subset {\mathbb Z}$ be a finite subset. We denote by $\mathcal{B}(A)$ the set of all integers $n \ge 2$ such that $|nA| > (2n-1)(|A|-1)$, where $nA=A+\cdots+A$ denotes the $n$-fold sumset of $A$. The motivation to consider…

The study of pinnacle sets has been a recent area of interest in combinatorics. Given a permutation, its pinnacle set is the set of all values larger than the values on either side of it. Largely inspired by conjectures posed by Davis,…

We consider the problem of finding $A_2(n,\{d_1,d_2\})$ defined as the maximal size of a binary (non-linear) code of length $n$ with two distances $d_1$ and $d_2$. Binary codes with distances $d$ and $d+2$ of size…

In order to investigate multiplicative structures in additively large sets, Beiglb\"{o}ck et al. raised a significant open question as to whether or not every subset of the natural numbers with bounded gaps (syndetic set) contains…

We say that a pair $(\mathcal{A},\mathcal{B})$ is a recovering pair if $\mathcal{A}$ and $\mathcal{B}$ are set systems on an $n$ element ground set, such that for every $A,A' \in \mathcal{A}$ and $B,B' \in \mathcal{B}$ we have that ($A…