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pro vyhledávání: '"52C10"'
Let $P$ be a $2n$-point set in the plane that is in general position. We prove that every red-blue bipartition of $P$ into $R$ and $B$ with $|R| = |B| = n$ generates $\Omega(n^{3/2})$ red-red-blue empty triangles.
Comment: 8 pages, 3 figures
Comment: 8 pages, 3 figures
Externí odkaz:
http://arxiv.org/abs/2409.17078
Autor:
Ranđelović, Žarko
Given functions $f,g: [n] \rightarrow [n]$ do there exist $n$ points $A_1,A_2\ldots A_n$ in some metric space such that $A_{f(i)},A_{g(i)}$ are the points closest and farthest from point $A_i$? In this paper we characterize precisely which pairs of f
Externí odkaz:
http://arxiv.org/abs/2409.14648
We study questions inspired by Erd\H os' celebrated distance problems with dot products in lieu of distances, and for more than a single pair of points. In particular, we study point configurations present in large finite point sets in the plane that
Externí odkaz:
http://arxiv.org/abs/2409.09683
Autor:
Tao, Terence
We show that for any large $n$, there exists a set of $n$ points in the plane with $O(n^2/\sqrt{\log n})$ distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a question o
Externí odkaz:
http://arxiv.org/abs/2409.01343
Autor:
Ruhland, Helmut
In a recently published article by G. Ambrus et al. a new upper bound for the density of an unit avoiding, periodic set is given as $0.2470$, the first upper bound $< 1/4$. A construction of Croft 1967 gave a lower bound $\delta_C = 0.22936$ for the
Externí odkaz:
http://arxiv.org/abs/2408.10076
Autor:
Cheek, Timothy, Cooper, Joseph, Gilman, Pico, Iosevich, Alex, Jaber, Kareem, Palsson, Eyvindur, Sharan, Vismay, Shuffelton, Jenna, Tomé, Marie-Hélène
We study a generalization of the Erd\H{o}s-Falconer distance problem over finite fields. For a graph $G$, two embeddings $p, p': V(G) \to \mathbb{F}_q^d$ of a graph $G$ are congruent if for all edges $(v_i, v_j)$ of $G$ we have that $||p(v_i) - p(v_j
Externí odkaz:
http://arxiv.org/abs/2408.07912
Autor:
Bezdek, Károly, Lángi, Zsolt
A problem of Erd\H{o}s (Amer. Math. Monthly 52: 494-498, 1945) and a theorem of Fejes T\'oth and Fejes T\'oth (Acta Math. Acad. Sci. Hungar. 24: 229-232, 1973) initiated the study of non-separable arrangements of convex bodies and the investigation o
Externí odkaz:
http://arxiv.org/abs/2407.20169
Autor:
Fraser, Jonathan M.
The Fourier transform plays a central role in many geometric and combinatorial problems cast in vector spaces over finite fields. In general, sets with good uniform bounds for the Fourier transform are less structured, more `random', and can often be
Externí odkaz:
http://arxiv.org/abs/2407.08589
Autor:
Bhowmick, Krishnendu
An old question posed by Erd\H{o}s asked whether there exists a set of $n$ points such that $c \cdot n$ distances occur more than $n$ times. We provide an affirmative answer to this question, showing that there exists a set of $n$ points such that $\
Externí odkaz:
http://arxiv.org/abs/2407.01174
For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the ordered Yao graph is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for every set of $n$ points
Externí odkaz:
http://arxiv.org/abs/2406.08913