Related papers: No-$(k+1)$-in-line problem for large constant $k$

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ edges on at most $s$ vertices. Brown, Erd\H{o}s and S\'os conjectured in 1973 that the limit $\lim_{n\rightarrow…

The classical no-three-in-line problem is to find the maximum number of points that can be placed in the $n \times n$ grid so that no three points lie on a line. Given a set $S$ of points in an Euclidean plane, the General Position Subset…

We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of…

Nielsen proved that the maximum number of maximal independent sets (MIS's) of size $k$ in an $n$-vertex graph is asymptotic to $(n/k)^k$, with the extremal construction a disjoint union of $k$ cliques with sizes as close to $n/k$ as…

We show that the lines of every arrangement of $n$ lines in the plane can be colored with $O(\sqrt{n/ \log n})$ colors such that no face of the arrangement is monochromatic. This improves a bound of Bose et al. \cite{BCC12} by a…

Let $n, r, k$ be positive integers such that $3\leq k < n$ and $2\leq r \leq k-1$. Let $m(n, r, k)$ denote the maximum number of edges an $r$-uniform hypergraph on $n$ vertices can have under the condition that any collection of $i$ edges,…

An $(n,k)$-Sperner partition system is a set of partitions of some $n$-set such that each partition has $k$ nonempty parts and no part in any partition is a subset of a part in a different partition. The maximum number of partitions in an…

We consider a minimizing variant of the well-known \emph{No-Three-In-Line Problem}, the \emph{Geometric Dominating Set Problem}: What is the smallest number of points in an $n\times n$~grid such that every grid point lies on a common line…

It is well known that an $N$-parameter $d$-dimensional Brownian sheet has no $k$-multiple points when $(k-1)d>2kN$, and does have such points when $(k-1)d<2kN$. We complete the study of the existence of $k$-multiple points by showing that…

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the 1-dimensional curve bounding a disk. For any nonnegative integer, a circle is called $n$-enclosing if…

For all integers $k\geq 3$, we give an $O(n^4)$ time algorithm for the problem whose instance is a graph $G$ of girth at least $k$ together with $k$ vertices and whose question is "Does $G$ contains an induced subgraph containing the $k$…

A subset $A$ of the $k$-dimensional grid $\{1,2, \cdots, N\}^k$ is called $k$-dimensional corner-free if it does not contain a set of points of the form $\{ a \} \cup \{ a + de_i : 1 \leq i \leq k \}$ for some $a \in \{1,2, \cdots, N\}^k$…

Given a word $w$, what is the maximum possible number of appearances of $w$ reading contiguously along any of the directions in $\{-1, 0, 1\}^d \setminus \{\mathbf{0}\}$ in a large $d$-dimensional grid (as in a word search)? Patchell and…

Erd\H{o}s asked what is the maximum number $\alpha(n)$ such that every set of $n$ points in the plane with no four on a line contains $\alpha(n)$ points in general position. We consider variants of this question for $d$-dimensional point…

We prove that for all positive integers $t$, every $n$-vertex graph with no $K_t$-subdivision has at most $2^{50t}n$ cliques. We also prove that asymptotically, such graphs contain at most $2^{(5+o(1))t}n$ cliques, where $o(1)$ tends to…

Finding the maximum number of induced cycles of length $k$ in a graph on $n$ vertices has been one of the most intriguing open problems of Extremal Graph Theory. Recently Balogh, Hu, Lidick\'{y} and Pfender answered the question in the case…

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph not containing a subhypergraph with $k$ edges on at most $s$ vertices. Recently, Delcourt and Postle, building on work of Glock, Joos, Kim,…

In 2023, Gollin, Hendrey, Methuku, Tompkins and Zhang determined the maximum number of cliques in general 1-planar graphs with order $n$. Their extremal examples have connectivity at most three, except for a few small orders. At the…

Let $S$ be a set of $n$ points in $\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k…

In this paper we show that at most $2 \gcd(m,n)$ points can be placed with no three in a line on an $m\times n$ discrete torus. In the situation when $\gcd(m,n)$ is a prime, we completely solve the problem.