Related papers: Greedy capsets
A cap set in projective or affine geometry over a finite field is a set of points no three of which are collinear. In this paper, we propose a new construction for complete cap sets that yields a cap set of size 124928 in the affine…
A set $\mathcal S\subset \mathbb N$ is said to be a subset-sum-distinct or dissociated if all of its finite subsets have different sums. Alternately, an equivalent classification is if any equality of the form $$\sum_{s\in \mathcal S}…
A $t$-spanner of a graph is a subgraph that $t$-approximates pairwise distances. The greedy algorithm is one of the simplest and most well-studied algorithms for constructing a sparse spanner: it computes a $t$-spanner with $n^{1+O(1/t)}$…
Let $S$ be a finite set of geometric objects partitioned into classes or \emph{colors}. A subset $S'\subseteq S$ is said to be \emph{balanced} if $S'$ contains the same amount of elements of $S$ from each of the colors. We study several…
Graphings are special bounded-degree graphs on probability spaces, representing limits of graph sequences that are convergent in a local or local-global sense. We describe a procedure for turning the underlying space into a compact metric…
We answer three questions posed in a paper by Babson and Benjamini. They introduced a parameter $C_G$ for Cayley graphs $G$ that has significant application to percolation. For a minimal cutset of $G$ and a partition of this cutset into two…
The random greedy algorithm for constructing a large partial Steiner-Triple-System is defined as follows. Begin with a complete graph on $n$ vertices and proceed to remove the edges of triangles one at a time, where each triangle removed is…
This paper defines a generalized column subset selection problem which is concerned with the selection of a few columns from a source matrix A that best approximate the span of a target matrix B. The paper then proposes a fast greedy…
A spanner graph on a set of points in $R^d$ contains a shortest path between any pair of points with length at most a constant factor of their Euclidean distance. In this paper we investigate new models and aim to interpret why good…
Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$. We show that if $|R| < \frac{3}{2}n$ and $P…
Let $Q$ be a finite subgraph of the integer grid $G$ in the plane, and let $T$ be a set of pairs of distinct vertices in $G$, called `terminal pairs'. Escaping a subset $X\subset T\cap Q$ from $Q$ means finding edge disjoint paths from the…
The random greedy algorithm for constructing a large partial Steiner-Triple-System is defined as follows. We begin with a complete graph on $n$ vertices and proceed to remove the edges of triangles one at a time, where each triangle removed…
The change-making problem was recently extended to sets of positive integers not containing the element $1$, and from there to numerical semigroups. A greedy numerical semigroup is defined as a numerical semigroup where the greedy…
We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle --- three different subsets $A,B,C\subseteq [n]$ such that…
An s-cap n-flat is given by a set of s points, no three of which are on a common line, in an n-dimensional affine space over the field of three elements. The cap set problem in dimension n is: what is the maximum s such that there is an…
Working over the field of order 2 we consider those complete caps (maximal sets of points with no three collinear) which are disjoint from some codimension 2 subspace of projective space. We derive restrictive conditions which such a cap…
Given a set of points in the plane, the \textsc{General Position Subset Selection} problem is that of finding a maximum-size subset of points in general position, i.e., with no three points collinear. The problem is known to be ${\rm…
Let $H_n^{(3)}$ be a 3-uniform linear hypergraph, i.e. any two edges have at most one vertex common. A special hypergraph, {\em wicket}, is formed by three rows and two columns of a $3 \times 3$ point matrix. In this note, we give a new…
Several concept learning problems can be regarded as special cases of half-space separation in abstract closure systems over finite ground sets. For the typical scenario that the closure system is implicitly given via a closure operator, we…
An $m$-general set in $AG(n,q)$ is a set of points such that any subset of size $m$ is in general position. A $3$-general set is often called a capset. In this paper, we study the maximum size of an $m$-general set in $AG(n,q)$,…