Related papers: A better bound for ordinary triangles

We show that for $m$ points and $n$ lines in the real plane, the number of distinct distances between the points and the lines is $\Omega(m^{1/5}n^{3/5})$, as long as $m^{1/2}\le n\le m^2$. We also prove that for any $m$ points in the…

Let $ABC$ be an equilateral triangle. For certain triangles $T$ (the "tile") and certain $N$, it is possible to cut $ABC$ into $N$ copies of $T$. It is known that only certain shapes of $T$ are possible, but until now very little was known…

We give a formula for counting the triangles in a picture consisting of the three sides of a triangle and some cevians. This lets us prove statements that are claimed without proof in the Online Encyclopedia of Integer Sequences and some…

We prove that any set of points in $\mathbb{R}^d$, any three of which form an angle less than $\frac{\pi}{3} + c$, has size $(1+\Theta(c))^d$ for sufficiently small $c>0$. The proof is based on a refinement of an approach by Erd\H{o}s and…

Let $\Pi_q$ be an arbitrary finite projective plane of order $q$. A subset $S$ of its points is called saturating if any point outside $S$ is collinear with a pair of points from $S$. Applying probabilistic tools we improve the upper bound…

Let $P$ be a set of $n$ points in the plane, each point moving along a given trajectory. A {\em $k$-collinearity} is a pair $(L,t)$ of a line $L$ and a time $t$ such that $L$ contains at least $k$ points at time $t$, the points along $L$ do…

An oriented graph is a directed graph which can be obtained from a simple undirected graph by orienting its edges. In this paper we show that any oriented graph G on n vertices with minimum indegree and outdegree at least (1/2-o(1))n…

Starting with any nondegenerate triangle we can use a well defined interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show…

We first introduce a configuration of arbitrary isogonal conjugates related to a known property concerning the spiral center of two pairs of isogonal conjugates. We then consider a special case where two conics are tangent at exactly two…

The planar Tur\'{a}n number of a given graph $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges over all planar graphs on $n$ vertices that do not contain a copy of $H$ as a subgraph. Let $H_k$ be a friendship graph,…

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the minimum number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We find lower bounds linear in $n$ for…

A triangular form is defined to be an integer-valued quadratic polynomial of the form $a_1P_3(x_1)+a_2P_3(x_2)+\cdots+a_kP_3(x_k)$ where $a_i's$ are positive integers and $P_3(x)=x(x+1)/2$. A triangular form is called regular if it…

Let $P = A\times A \subset \mathbb{F}_p \times \mathbb{F}_p$, $p$ a prime. Assume that $P= A\times A$ has $n$ elements, $n<p$. See $P$ as a set of points in the plane over $\mathbb{F}_p$. We show that the pairs of points in $P$ determine…

The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_k$ denote the cycle of length $k$. The planar Tur\'an number…

The expansion $F^{\triangle}$ of a graph $F$ is the graph obtained from $F$ by replacing each edge with a triangle. Lv \etal proposed a conjecture on the maximum number of triangles in a graph without $P_k^{\triangle}$ or $C_k^{\triangle}$…

In this article we proof the existence of 4-regular planar unit-distance graphs consisting only of unit triangles without additional triangles. It is shown that the smallest number of unit triangles is $\leq$6422.

Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of $n$ points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$…

Let $p$ be a fixed prime. A triangle in $\mathbb{F}_p^n$ is an ordered triple $(x,y,z)$ of points satisfying $x+y+z=0$. Let $N=p^n=|\mathbb{F}_p^n|$. Green proved an arithmetic triangle removal lemma which says that for every $\epsilon>0$…

A drawing of a graph is $k$-plane if every edge contains at most $k$ crossings. A $k$-plane drawing is saturated if we cannot add any edge so that the drawing remains $k$-plane. It is well-known that saturated $0$-plane drawings, that is,…

A topological graph drawn on a cylinder whose base is horizontal is \emph{angularly monotone} if every vertical line intersects every edge at most once. Let $c(n)$ denote the maximum number $c$ such that every simple angularly monotone…