Andreas Holmsen, Some recent results on geometric transversals

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 and old results and problems regarding geometric transversals, based on joint work with Otfried Cheong and Xavier Goaoc.

Andreas Holmsen, Discrete geometry in convexity spaces

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, Fractional Helly and topological complexity

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 of subsets of $\mathbb R^d$ which satisfy only very weak topological assumptions. The proofs combine a number of tools such as homological minors, stair-convexity, supersaturation in hypergraphs, Radon dimension, and Ramsey-type arguments. This is joint work with Xavier Goaoc and Zuzana Patáková.

Andreas Holmsen presented his work on an extremal problem on hypergraphs on March 12 at the discrete math seminar

Andreas Holmsen from KAIST gave a talk at Discrete Math Seminar on March 12, 2019 under the title “large cliques in hypergraphs with forbidden substructures.” His work extended a result of extremal graph theory due to Gyárfás, Hubenko, and Solymosi, answering a question of Erdős to hypergraphs and has interesting consequences in topological combinatorics and abstract convexity.

Andreas Holmsen, Large cliques in hypergraphs with forbidden substructures

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 of an induced $K_{2,2}$, which allows us to extend their result to $k$-uniform hypergraphs. Our result also has interesting consequences in topological combinatorics and abstract convexity, where it can be used to answer questions by Bukh, Kalai, and several others.