Related papers: Maximal antichains of subsets II: Constructions

Let $\{(A_i,B_i)\}_{i=1}^m$ be a set pair system. F\"{u}redi, Gy\'{a}rf\'{a}s and Kir\'{a}ly called it {\em $1$-cross intersecting} if $|A_i\cap B_j|$ is $1$ when $i\neq j$ and $0$ if $i=j$. They studied such systems and their…

A critical set in an $n \times n$ array is a set $C$ of given entries, such that there exists a unique extension of $C$ to an $n\times n$ Latin square and no proper subset of $C$ has this property. The cardinality of the largest critical…

We study the smallest, as well as the largest numbers of congruences of lattices of an arbitrary finite cardinality $n$. Continuing the work of Freese and Cz\' edli, we prove that the third, fourth and fifth largest numbers of congruences…

The construction of complementary sets (CSs) of sequences with different set size and sequence length become important due to its practical application for OFDM systems. Most of the constructions of CSs, based on generalized Boolean…

Given integer $n > 0$ and $m > 1$, we call a partition of set $[n] = \{1, \dots, n\}$ {\em $m$-good} if each of the partitioning sets is of size at most $m$ and the sum of numbers in it is a power of $m$, that is, $m^t$ for some $t \geq 0$.…

In this paper, we study minimal (with respect to inclusion) zero forcing sets. We first investigate when a graph can have polynomially or exponentially many distinct minimal zero forcing sets. We also study the maximum size of a minimal…

A $\left(n,\ell,\gamma\right)$-sharing set family of size $m$ is a family of sets $S_1,\ldots,S_m\subseteq [n]$ s.t. each set has size $\ell$ and each pair of sets shares at most $\gamma$ elements. We let $m\left(n,\ell,\gamma\right)$…

A binary poset code of codimension M (of cardinality 2^{N-M}, where N is the code length) can correct maximum M errors. All possible poset metrics that allow codes of codimension M to be M-, (M-1)- or (M-2)-perfect are described. Some…

The highest possible minimal norm of a unimodular lattice is determined in dimensions n <= 33. There are precisely five odd 32-dimensional lattices with the highest possible minimal norm (compared with more than 8*10^20 in dimension 33).…

For given posets $P$ and $Q$ and an integer $n$, the generalized Tur\'an problem for posets, asks for the maximum number of copies of $Q$ in a $P$-free subset of the $n$-dimensional Boolean lattice, $2^{[n]}$. In this paper, among other…

We study the typical behavior of the least common multiple of the elements of a random subset $A\subset \{1,\dots, n\}$. For example we prove that $\text{lcm}\{a:\ a\in A\}=2^{n(1+o(1))}$ for almost all subsets $A\subset\{1,\dots,n\}$.

Let $k$, $t$ and $m$ be positive integers. A $k$-multiset of $[m]$ is a collection of $k$ integers from the set $\{1,...,m\}$ in which the integers can appear more than once. We use graph homomorphisms and existing theorems for intersecting…

Given $A$ a set of $N$ positive integers, an old question in additive combinatorics asks that whether $A$ contains a sum-free subset of size at least $N/3+\omega(N)$ for some increasing unbounded function $\omega$. The question is generally…

We find nearly the optimal size of a set $A\subset [N] := \{1,...,N\}$ so that the product set $AA$ satisfies either (i) $|AA| \sim |A|^2/2$ or (ii) $|AA| \sim |[N][N]|$. This settles problems raised in a recent article of Cilleruelo,…

Let $G$ be a finite group. A family $\mathcal{M}$ of maximal subgroups of $G$ is called `irredundant' if its intersection is not equal to the intersection of any proper subfamily. $\mathcal{M}$ is called `maximal irredundant' if…

An error model with asymmetric single magnitude four error is considered. This paper is about constructions of codes correcting single error over $\mathbb{Z}_{2^{a}3^{b}r}$. Firstly, we reduce the construction of a maximal size…

Let $M$ be a perfect matching in a graph. A subset $S$ of $M$ is said to be a forcing set of $M$, if $M$ is the only perfect matching in the graph that contains $S$. The minimum size of a forcing set of $M$ is called the forcing number of…

Let $L_{mkl}\subset \mathbb{R}^{m+k+l}$ be the set of vectors which have $m$ of entries $-1$, $k$ of entries $0$, and $l$ of entries $1$. In this paper, we investigate the largest subset of $L_{mkl}$ whose diameter is smaller than that of…

Let $\mathcal{T}$ be a collection of 3-element subsets $S$ of $\{1, \ldots,n\}$ with the property that if $i<j<k$ and $a<b<c$ are two 3-element subsets in $S$, then there exists an integer sequence $x_1 < x_2 < \cdots < x_n$ such that $x_i,…

In this paper, we investigate a problem concerning quartets, which are a particular type of tree on four leaves. Loosely speaking, a set of quartets is said to be `definitive' if it completely encapsulates the structure of some larger tree,…