On May 16, 2022, Andreas Holmsen from KAIST presented a theorem proving a colorful generalization of the Goodman-Pollack-Wenger theorem, answering a conjecture of Arocha, Bracho, and Montejano, at the Discrete Math Seminar. The title of his talk was “A colorful version of the Goodman-Pollack-Wenger transversal theorem“.

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

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 Arocha, Bracho, and Montejano. The proof uses topological methods, in particular the Borsuk-Ulam theorem. The same methods also allow us to generalize some colorful transversal theorems of Montejano and Karasev.

## Andreas Holmsen gave a talk about a counterexample to the Bárány-Kalai conjecture on line transversals of a finite family of convex sets at the Discrete Math Seminar

On January 11, 2022, Andreas Holmsen from KAIST gave a talk at the Discrete Math Seminar about the counterexample to the Bárány-Kalai conjecture on line transversals of a finite family of convex sets. The title of his talk was “Some recent results on geometric transversals“.

## 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 gave a talk on combinatorial geometry in convexity spaces at the Discrete Math Seminar

On January 12, 2021, Andreas Holmsen from KAIST gave a talk on the combinatorial geometry in convexity spaces. The title of his talk was “Discrete geometry in convexity spaces“.

This seminar talk was in honor of Helge Tverberg, who passed away in December 28, 2020 and was the Ph.D. advisor of Andreas Holmsen.

## 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 gave a talk on a recent work generalizing the fractional Helly theorem under very weak topological assumptions at the Discrete Math Seminar

On June 16, 2020, Andreas Holmsen from KAIST presented his recent work with Xavier Goaoc and Zuzana Patáková on generalizing the fractional Halley theorem under very weak topological assumptions. The title of his talk was “Fractional Helly and topological complexity“.

## 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.