Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPoint sets with no four collinear and no large visible island
237
0
0.0
(
0
)
Added by
Andrew Suk
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Given a finite point set $P$ in the plane, a subset $S subseteq P$ is called an island in $P$ if $conv(S) cap P = S$. We say that $Ssubset P$ is a visible island if the points in $S$ are pairwise visible and $S$ is an island in $P$. The famous Big-line Big-clique Conjecture states that for any $k geq 3$ and $ell geq 4$, there is an integer $n = n(k,ell)$, such that every finite set of at least $n$ points in the plane contains $ell$ collinear points or $k$ pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by constructing arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size $2^{42}$.
rate research
Read More
We prove the following generalised empty pentagon theorem: for every integer $ell geq 2$, every sufficiently large set of points in the plane contains $ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the big line or big clique conjecture of Kara, Por, and Wood [emph{Discrete Comput. Geom.} 34(3):497--506, 2005].
Given a finite set $A subseteq mathbb{R}^d$, points $a_1,a_2,dotsc,a_{ell} in A$ form an $ell$-hole in $A$ if they are the vertices of a convex polytope which contains no points of $A$ in its interior. We construct arbitrarily large point sets in general position in $mathbb{R}^d$ having no holes of size $O(4^ddlog d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as $(t,m,s)$-nets or $(t,s)$-sequences, yielding a bound of $2^{7d}$. The better bound is obtained using a variant of $(t,m,s)$-nets, obeying a relaxed equidistribution condition.
We will prove that there exists a model of ZFC+``c= omega_2 in which every M subseteq R of cardinality less than continuum c is meager, and such that for every X subseteq R of cardinality c there exists a continuous function f:R-> R with f[X]=[0,1]. In particular in this model there is no magic set, i.e., a set M subseteq R such that the equation f[M]=g[M] implies f=g for every continuous nowhere constant functions f,g:R-> R .
Let $W$ be a compact smooth $4$-manifold that deformation retract to a PL embedded closed surface. One can arrange the embedding to have at most one non-locally-flat point, and near the point the topology of the embedding is encoded in the singularity knot $K$. If $K$ is slice, then $W$ has a smooth spine, i.e., deformation retracts onto a smoothly embedded surface. Using the obstructions from the Heegaard Floer homology and the high-dimensional surgery theory, we show that $W$ has no smooth spines if $K$ is a knot with nonzero Arf invariant, a nontrivial L-space knot, the connected sum of nontrivial L-space knots, or an alternating knot of signature $<-4$. We also discuss examples where the interior of $W$ is negatively curved.
A word is square-free if it does not contain any square (a word of the form $XX$), and is extremal square-free if it cannot be extended to a new square-free word by inserting a single letter at any position. Grytczuk, Kordulewski, and Niewiadomski proved that there exist infinitely many ternary extremal square-free words. We establish that there are no extremal square-free words over any alphabet of size at least 17.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
