Andreas Holmsen, Large cliques in hypergraphs with forbidden substructures

Room B232 IBS (기초과학연구원)

A result due to Gyárfás, Hubenko, and Solymosi, answering a question of Erdős, asserts that if a graph $G$ does not contain $K_{2,2}$ as an induced subgraph yet has at least $c\binom{n}{2}$ edges, then $G$ has a complete subgraph on at least $\frac{c^2}{10}n$ vertices. In this paper we suggest a "higher-dimensional" analogue of the notion

Andreas Holmsen, Fractional Helly and topological complexity

Room B232 IBS (기초과학연구원)

The fractional Helly theorem is a simple yet remarkable generalization of Helly's classical theorem on the intersection of convex sets, and it is of considerable interest to extend the fractional Helly theorem beyond the setting of convexity. In this talk I will discuss a recent result which shows that the fractional Helly theorem holds for families

Andreas Holmsen, Discrete geometry in convexity spaces

Room B232 IBS (기초과학연구원)

The notion of convexity spaces provides a purely combinatorial framework for certain problems in discrete geometry. In the last ten years, we have seen some progress on several open problems in the area, and in this talk, I will focus on the recent results relating to Tverberg’s theorem and the Alon-Kleitman (p,q) theorem.

Andreas Holmsen, Some recent results on geometric transversals

Room B232 IBS (기초과학연구원)

A geometric transversal to a family of convex sets in $\mathbb R^d$ is an affine flat that intersects the members of the family. While there exists a far-reaching theory concerning 0-dimensional transversals (intersection patterns of convex sets), much less is known when it comes to higher-dimensional transversals. In this talk, I will present some new

Andreas Holmsen, A colorful version of the Goodman-Pollack-Wenger transversal theorem

Room B332 IBS (기초과학연구원)

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 in $\mathbb{R}^d$ by Goodman, Pollack, and Wenger. Here we establish a colorful extension of their theorem, which proves a conjecture of

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