Related papers: Optimally Covering Large Triangles with Homothetic…
Conway and Soifer showed that an equilateral triangle $T$ of side $n + \varepsilon$ with sufficiently small $\varepsilon > 0$ can be covered by $n^2 + 2$ unit equilateral triangles. They conjectured that it is impossible to cover $T$ with…
In a famous short paper, Conway and Soifer show that $n^2 + 2$ equilateral triangles with edge length 1 can cover one with side $n + \epsilon$. We provide a generalization to $d$ dimensions.
This work revolves around the two following questions: Given a convex body $C\subset\mathbb{R}^d$, a positive integer $k$ and a finite set $S\subset\mathbb{R}^d$ (or a finite Borel measure $\mu$ on $\mathbb{R}^d$), how many homothets of $C$…
Let $H$ be an $n$-vertex 3-uniform hypergraph such that every pair of vertices is in at least $n/3+o(n)$ edges. We show that $H$ contains two vertex-disjoint tight paths whose union covers the vertex set of $H$. The quantity two here is…
The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n)<=M_{ubt}(d,n) provided by…
Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with $n$ vertices and maximum degree at most $r$, where $n =…
Let $\mathcal{T}$ be a finite nonempty set of $3$-element subsets of a totally ordered set $V$. We view $\mathcal{T}$ as the set of triangles in the support graph. Let $\delta_{1,\mathcal{T}}$ be the signed edge-triangle incidence matrix,…
The widely studied edge modification problems ask how to minimally alter a graph to satisfy certain structural properties. In this paper, we introduce and study a new edge modification problem centered around transforming a given graph into…
Recently, Ning and Zhai (2023) proved that every $n$-vertex graph $G$ with $\lambda (G) \ge \sqrt{\lfloor n^2/4\rfloor}$ has at least $\lfloor n/2\rfloor -1$ triangles, unless $G=K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}$.…
We investigate a covering problem in $3$-uniform hypergraphs ($3$-graphs): given a $3$-graph $F$, what is $c_1(n,F)$, the least integer $d$ such that if $G$ is an $n$-vertex $3$-graph with minimum vertex degree $\delta_1(G)>d$ then every…
Let $H$ be a $k$-uniform $D$-regular simple hypergraph on $N$ vertices. Based on an analysis of the R\"odl nibble, Alon, Kim and Spencer (1997) proved that if $k \ge 3$, then $H$ contains a matching covering all but at most…
Fix an integer n>=1. Suppose that a simple polygon is the union of n triangles whose vertices along the common boundary are arranged cyclically. How many sides can such a union -- to be called regular -- have at most? This gives OEIS…
We revisit the classic Maximum $k$-Coverage problem: Determine the largest number $t$ of elements that can be covered by choosing $k$ sets from a given family $\mathcal{F} = \{S_1,\dots, S_n\}$ of a size-$u$ universe. A notable special case…
More than forty years ago, Erd\H{o}s conjectured that for any T <= N/K, every K-uniform hypergraph on N vertices without T disjoint edges has at most max{\binom{KT-1}{K}, \binom{N}{K} - \binom{N-T+1}{K}} edges. Although this appears to be a…
In this paper we present a new bound obtained with the probabilistic method for the solution of the Set Covering problem with unit costs. The bound is valid for problems of fixed dimension, thus extending previous similar asymptotic…
The study of extremal problems on triangle areas was initiated in a series of papers by Erd\H{o}s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that…
This paper introduces path triangulation of points in a bounded, simply connected surface region, replacing ordinary triangles in a Delaunay triangulation with path triangles from homotopy theory. A {\bf path triangle} has a border that is…
We study the problems of bounding the number weak and strong independent sets in $r$-uniform, $d$-regular, $n$-vertex linear hypergraphs with no cross-edges. In the case of weak independent sets, we provide an upper bound that is tight up…
An edge clique cover of a graph is a set of cliques that covers all edges of the graph. We generalize this concept to "$K_t$ clique cover", i.e. a set of cliques that covers all complete subgraphs on $t$ vertices of the graph, for every $t…
In this paper, we investigate the homothetic point enclosure problem: given a set $S$ of $n$ triangles with sides parallel to three fixed directions, find a data structure for $S$ that can report all the triangles of $S$ that contain a…