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For a rational number $r>1$, a set $A$ of positive integers is called an $r$-multiple-free set if $A$ does not contain any solution of the equation $rx = y$. The extremal problem on estimating the maximum possible size of $r$-multiple-free…
We consider a problem of maximizing the product of the sizes of two uniform cross-$t$-intersecting families of sets. We show that the value of this maximum is at most polynomially larger (in the size of a ground set) than a quantity…
An $(n,k)$-Sperner partition system is a collection of partitions of some $n$-set, each into $k$ nonempty classes, such that no class of any partition is a subset of a class of any other. The maximum number of partitions in an…
An s-cap n-flat, or an n-dimensional cap of size s, is a pair (S,F) where F is an n-dimensional affine space over Z/3Z and the size-s subset S of F contains no triple of collinear points. The cap set problem in dimension n asks for the…
We prove that there are arbitrarily large indecomposable ordered sets T with a 2-chain C such that the smallest indecomposable proper superset U of C in T is T itself. Subsequently, we characterize all such indecomposable ordered sets T and…
This is an overview about a method of constructing ccc forcings: Suppose first that a continuous, commutative system of complete embeddings between countable forcings indexed along $\omega_1$ is given. Then its direct limit satisfies ccc by…
Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter---the total length of all pieces of their boundaries visible from above. We prove that if the centers…
This paper considers coresets for the robust $k$-medians problem with $m$ outliers, and new constructions in various metric spaces are obtained. Specifically, for metric spaces with a bounded VC or doubling dimension $d$, the coreset size…
We show that certain families of sets in $\mathbb{R}^2$ (or $\mathbb{R}^n$) which are neither definable nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure.
Visualizations of set systems frequently use enclosing geometries for the sets in combination with reduced representations of the elements, such as short text labels, small glyphs, or points. Hence they are generally unable to adequately…
A subset $\mathcal{C}\subseteq\{0,1,2\}^n$ is said to be a $\textit{trifferent}$ code (of block length $n$) if for every three distinct codewords $x,y, z \in \mathcal{C}$, there is a coordinate $i\in \{1,2,\ldots,n\}$ where they all differ,…
In this paper, we explore the concept of Mutually Unbiased Bases (MUBs) in discrete quantum systems. It is known that for dimensions $d$ that are powers of prime numbers, there exists a set of up to $d+1$ bases that form an MUB set.…
A subset $A$ of the integers is a $B_k[g]$ set if the number of multisets from $A$ that sum to any fixed integer is at most $g$. Let $F_{k,g}(n)$ denote the maximum size of a $B_k[g]$ set in $\{1,\dots, n\}$. In this paper we improve the…
For a point set of $n$ elements in the $d$-dimensional unit cube and a class of test sets we are interested in the largest volume of a test set which does not contain any point. For all natural numbers $n$, $d$ and under the assumption of a…
This paper shows how maximum possible configuration efficiency of an indefinitely large software system is constrained by chosing a fixed upper limit to the number of program units per subsystem. It is then shown how the configuration…
Let $A \subset \mathbb{Z}_{>0}$ of size $n$. It is conjectured that for any $C >0$ and $n$ large enough that $A$ contains a sum-free subset of size at least $n/3 +C$. We study this problem and find an alternate proof of Bourgain's result…
We give a complete classification of globally generated vector bundles of rank 3 on a smooth quadric threefold with $c_1\leq 2$ and extend the result to arbitrary higher rank case. We also investigate the existence of globally generated…
Given $n+1$ unit vectors in $\mathbf{R}^n$ or $\mathbf{C}^n,$ consider the absolute values of the determinants of the vectors taken $n$ at a time. By taking a geometric perspective, we show that the minimum of these determinants is…
Let F be a family of subsets of an n-element set not containing four distinct members such that A union B is contained in C intersect D. It is proved that the maximum size of F under this condition is equal to the sum of the two largest…
This paper takes a computational learning theory approach to a problem of linear systems identification. It is assumed that input signals have only a finite number k of frequency components, and systems to be identified have dimension no…