Reading Seminar of the Combinatorial Geometry Research Group

2025 Spring Semester - every Friday at 10:30 in the Turán Room, Rényi

Preliminary schedule:

May 16, 2025
Presentation of theses
Miklós Csizmadia: Improvement on line transversals of families of connected sets on the plane

Abstract: Three lines are concurrent if they intersect at a single point. In this paper I prove that if $F$ is a bounded family of compact connected sets in the plane, such that every three sets in $F$ can be pierced by a single line, then there exists three concurrent lines in the plane such that the union of the three lines intersect every member of $F$. This had previously only been proven for lines that are not required to be concurrent by McGinnis and Zerbib.

Máté Jánosik: Arrangements of pseudocircles

Abstract: Arrangements of curves have long played a central role in discrete and computational geometry. In particular, the study of pseudocircles—simple closed curves in the plane or on the sphere that pairwise intersect exactly twice—offers a powerful generalization of both arrangements of straight lines and arrangements of true circles. This topic was pioneered by Branko Grünbaum in the 1970s, who posed fundamental questions about the combinatorial structure of such arrangements: How many $k$-gonal faces (in particular, triangles and digons) can appear? Which arrangements can actually be realized by proper circles? And how do classical incidence theorems extend to this broader setting? I will talk about the recent progress made on the number of digons and triangles in pairwise intersecting arrangements.

Domonkos Vizsy: Erdős–Szekeres-type theorems for monotone paths

Abstract: An ordered graph is a graph together with a total ordering of its vertices. Let $H$ be an ordered $k$-uniform hypergraph. We say that $H$ contains a monotone path of length $n$ if there is an ordered sequence of vertices of length $n+k-1$ for which $H$ contains every $k$-tuple (i.e. edge) of consecutive vertices in the sequence. Let $N_k(q,n)$ be the smallest integer $N$ so that every $q$-coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic monotone path of length $n$. The problem of bounding $N_k(q,n)$ was studied by Fox, Pach, Sudakov and Suk; however, it goes back (implicitly) to the seminal 1935 paper of Erdős and Szekeres. In this talk, I present a construction by Moshkovitz and Shapira that improves the bound of Fox et al.; furthermore, I investigate how $N_k(q,n)$ changes if the coloring is restricted to stricter properties, and I present a construction by Balko addressing this case.


May 9, 2025
Pázmány Day, no seminar

May 2, 2025
Holiday, no seminar

April 25, 2025
József Osztényi: Tight bounds for intersection-reverse sequences, edge-ordered graphs and applications

Bibliography:
Barnabás Janzer, Oliver Janzer, Abhishek Methuku, Gábor Tardos: Tight bounds for intersection-reverse sequences, edge-ordered graphs and applications, arXiv


April 18, 2025
Spring break, no seminar

April 11, 2025
Péter Ágoston: Many antipodes implies many neighbors

Bibliography:
Stefan Steinerberger: Many antipodes implies many neighbors, arXiv


April 4, 2025
Miklós Csizmadia: Colorful Helly via induced matchings

Bibliography:
Cosmin Pohoata, Kevin Yang, Shengtong Zhang: Colorful Helly via induced matchings, arXiv


March 28, 2025
Máté Jánosik: Chasing puppies on orthogonal straight-line plane graphs

Bibliography:
Johanna Ockenfels, Yoshio Okamoto, Patrick Schnider: Chasing puppies on orthogonal straight-line plane graphs, arXiv


March 21, 2025
Rebeka Raffay: Non-Euclidean Erdős-Anning Theorems

Bibliography:
David Eppstein: Non-Euclidean Erdős-Anning Theorems, arXiv


March 14, 2025
Dömötör Pálvölgyi: Polynomial Gyárfás-Sumner conjecture for graphs of bounded boxicity

Bibliography:
James Davies, Yelena Yuditsky: Polynomial Gyárfás-Sumner conjecture for graphs of bounded boxicity, arXiv


March 7, 2025
Lili Ködmön: Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs

Bibliography:
Aaron Büngener, Michael Kaufmann: Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs, arXiv


February 28, 2025
Attila Jung: The fractional Helly number for separable convexity spaces

Abstract: I present a result of Holmsen and Patáková about fractional Helly numbers of separable convexity spaces, which generalizes a result of Bárány and Matoušek about the fractional Helly number of convex lattice sets.

Bibliography:
Andreas F. Holmsen, Zuzana Patáková: The fractional Helly number for separable convexity spaces, arXiv