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Let $C \subseteq [r]^m$ be a code such that any two words of $C$ have Hamming distance at least $t$. It is not difficult to see that determining a code $C$ with the maximum number of words is equivalent to finding the largest $n$ such that…
Best-Fit is one of the most prominent and practically used algorithms for the bin packing problem, where a set of items with associated sizes needs to be packed in the minimum number of unit-capacity bins. Kenyon [SODA '96] studied online…
We consider an effective theory of massive scalar fields on a fixed AdS$_{d+1}$ background with a cubic extremal interaction among them. A bulk coupling is called extremal whenever the corresponding conformal dimension of any of the dual…
We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the…
While well-known methods to list the intersections of either a list of segments or a complex polygon aim at achieving optimal time-complexity they often do so at the cost of memory comsumption and complex code. Real-life software…
A $2k$-matching is a perfect matching of the complete graph on $2k$ vertices. Two $2k$-matchings are defined to be $t$-intersecting if they have at least $t$ edges in common. The main result in this paper is that if $k \geq 3t/2+1$, then…
We prove three theorems giving extremal bounds on the incidence structures determined by subsets of the points and blocks of a balanced incomplete block design (BIBD). These results generalize and strengthen known bounds on the number of…
Given a complete graph with $n$ vertices and non-negative edge weights, where $n$ is divisible by 3, the maximum weight 3-path packing problem is to find a set of $n/3$ vertex-disjoint 3-paths such that the total weight of the 3-paths in…
In this note we address the problem of determining the maximum number of points of intersection of two arithmetically Cohen-Macaulay curves in $\PP^3$. We give a sharp upper bound for the maximum number of points of intersection of two…
We consider problems of finding a maximum size/weight $t$-matching without forbidden subgraphs in an undirected graph $G$ with the maximum degree bounded by $t+1$, where $t$ is an integer greater than $2$. Depending on the variant forbidden…
The cyclic code is a subclass of linear codes and has applications in consumer electronics, data storage systems and communication systems due to the efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open…
For any $\varepsilon > 0$, we prove that $k$-Dimensional Matching is hard to approximate within a factor of $k/(12 + \varepsilon)$ for large $k$ unless $\textsf{NP} \subseteq \textsf{BPP}$. Listed in Karp's 21 $\textsf{NP}$-complete…
A family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is called $k$-wise intersecting if any $k$ members of $\mathcal{F}$ have non-empty intersection, and it is called maximal $k$-wise intersecting if no family strictly containing…
The maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=6$, and constant dimension $k=4$ is $257$, where the $2$ isomorphism types are extended lifted maximum rank distance codes. In…
In the Colored Bin Packing problem a sequence of items of sizes up to $1$ arrives to be packed into bins of unit capacity. Each item has one of $c\geq 2$ colors and an additional constraint is that we cannot pack two items of the same color…
The maximum size of $t$-intersecting families is one of the most celebrated topics in combinatorics, and its size is known as the Erd\H{o}s-Ko-Rado theorem. Such intersecting families, also known as constant-weight anticodes in coding…
Guo and Xu determined the maximum size of intersecting families over finite affine spaces and showed that any family reaches maximum size must be trivial. In this paper, we characterize non-trivial intersecting family with maximum size.
In this note, we first bound the intersection number of the regular simplicial partitions.
We study the intersection of tropical psi-classes on tropical heavy/light Hassett spaces, generalising a result of Kerber--Markwig for tropical moduli spaces of rational stable curves with distinct marked points. Our computation reveals…
Consider a problem where we are given a bipartite graph H with vertices arranged on two horizontal lines in the plane, such that the two sets of vertices placed on the two lines form a bipartition of H. We additionally require that H admits…