Zobrazeno 1 - 10
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pro vyhledávání: '"Holmsen, Andreas"'
We prove a generalization of the topological Tverberg theorem. One special instance of our general theorem is the following: Let $\Delta$ denote the 8-dimensional simplex viewed as an abstract simplicial complex, and suppose that its vertices are arr
Externí odkaz:
http://arxiv.org/abs/2412.16055
Autor:
Holmsen, Andreas F., Patáková, Zuzana
A convex lattice set in $\mathbb{Z}^d$ is the intersection of a convex set in $\mathbb{R}^d$ and the integer lattice $\mathbb{Z}^d$. A well-known theorem of Doignon states that the Helly number of $d$-dimensional convex lattice sets equals $2^d$, whi
Externí odkaz:
http://arxiv.org/abs/2412.01445
Autor:
Holmsen, Andreas F.
We report on some recent progress regarding combinatorial properties in convexity spaces with a bounded Radon number. In particular, we discuss the relationship between the Radon number, the colorful and fractional Helly properties, weak $\varepsilon
Externí odkaz:
http://arxiv.org/abs/2408.05871
The colorful Helly theorem and Tverberg's theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers $d \geq m \geq 1$ and $k$ a prime power. Supp
Externí odkaz:
http://arxiv.org/abs/2403.14909
A graph class $\mathcal{G}$ has the strong Erd\H{o}s-Hajnal property (SEH-property) if there is a constant $c=c(\mathcal{G}) > 0$ such that for every member $G$ of $\mathcal{G}$, either $G$ or its complement has $K_{m, m}$ as a subgraph where $m \geq
Externí odkaz:
http://arxiv.org/abs/2302.02417
We prove that for any set $F$ of $n\ge 2$ pairwise disjoint open convex sets in $\mathbb{R}^3$, the connected components of the set of lines intersecting every member of $F$ are contractible. The same result holds for directed lines.
Comment: Th
Comment: Th
Externí odkaz:
http://arxiv.org/abs/2205.14681
Autor:
Holmsen, Andreas F.
Hadwiger's transversal theorem gives necessary and sufficient conditions for the existence of a line transversal to a family of pairwise disjoint convex sets in the plane. These conditions were subsequently generalized to hyperplane transversals to g
Externí odkaz:
http://arxiv.org/abs/2205.04077
We prove that there exist no weak $\varepsilon$-nets of constant size for lines and convex sets in $\mathbb{R}^d$.
Comment: Boris Bukh has informed us that our construction can be extended to 2-flats in $\mathbb{R}^4$. The contents of this manus
Comment: Boris Bukh has informed us that our construction can be extended to 2-flats in $\mathbb{R}^4$. The contents of this manus
Externí odkaz:
http://arxiv.org/abs/2202.02719
Autor:
Holmsen, Andreas F.
We give a new proof of a theorem of Montejano and Karasev regarding $k$-dimensional transversals to small families of convex sets. While their proof uses technical algebraic and topological tools, our proof is a simple application of the Borsuk-Ulam
Externí odkaz:
http://arxiv.org/abs/2112.07907
In this paper we study generalizations of classical results on intersection patterns of set systems in $\mathbb{R}^d$, such as the fractional Helly theorem or the $(p,q)$-theorem, in the setting of arbitrary triangulable spaces with a forbidden homol
Externí odkaz:
http://arxiv.org/abs/2103.09286