Speaker: Rainie Heck (Bolyai Institute, University of Szeged and Eötvös University)

Title: Vector sum problems in convex and discrete geometry

Time: October 28, 2025 12:30 pm

Location: Riesz lecture hall, Bolyai Building, Aradi vértanúk tere 1., Szeged, Hungary

Abstract: In this talk we will focus on two problems from discrete and convex geometry: the vector balancing problem and the Steinitz problem. After introducing each problem and its history (including a surprising connection between them!), we present results for a generalization of vector balancing and for a reduction of the Steinitz problem. More precisely, we study a geometric generalization of the vector balancing problem called colorful vector balancing, and we show that two important results from the original problem also hold (and are tight and asymptoticaly tight, respectively) in the colorful setting as well. We also prove a reduction of the Steinitz problem to a more approachable setting, offering a potential proof avenue for a long standing open conjecture.

Speaker: Zsolt Lángi (University of Szeged and Rényi Institute)

Title: Arclength of curves with the increasing chords property

Time: October 21, 2025 12:30 pm

Location: Riesz lecture hall, Bolyai Building, Aradi vértanúk tere 1., Szeged, Hungary

Abstract: We say that a curve $\gamma$ satisfies the increasing chords property, if for any points a,b,c,d in this order on $\gamma$, the distance of a,d is not smaller than the distance of b,c. Binmore asked the question in 1971 if there is a universal constant C such that for any curve $\gamma$ in the Euclidean plane, satisfying the increasing chords property, if the endpoints of $\gamma$ are at unit distance apart, then the arclength of $\gamma$ is at most C. Larman and McMullen showed in 1972 that the constant $C=2\sqrt{3}$ satisfies this condition. Rote proved in 1991 that the optimal such constant is equal to $\frac{2\pi}{3}$. In this note we give an estimate for the arclengths of curves with the increasing chords property in Euclidean d-space, and generalize Rote's result for such curves in a normed plane with a strictly convex norm. Joint work with Adrian Dumitrescu and Sara Lengyel.

Speaker: Adrian Dumitrescu (Algoresearch L.L.C., Milwaukee, USA)

Title: Subset Selection Problems in Planar Point Sets

Time: October 14, 2025 12:30 pm

Location: Riesz lecture hall, Bolyai Building, Aradi vértanúk tere 1., Szeged, Hungary

Abstract: (I) Given a set of points in the plane, the General Position Subset Selection problem is that of finding a maximum-size subset of points in general position,
i.e., with no three points collinear. The problem is known to be NP-complete and APX-hard, and the best approximation ratio known is Omega(n^{-1/2}). Here we obtain better approximations in three specials cases; for example, we obtain a
Omega((\log{n})^{-1/2})-approximation for the case where the input set is the set of vertices of a generic n-line arrangement, i.e., one with Omega(n^2) vertices.

(II) We study variations of themes introduced / studied by
(a) Dudeney, (b) Erdős-Szekeres, (c) Erdős, Graham, Ruzsa, and Taylor,
(d) Gowers, (e) Payne-Wood, and (f) Zhang, from the combinatorial point of view. Given a set P of n points:

A. Find a largest general position subset, i.e., with no three collinear
B. Find a largest monotone general position subset
C. Find a largest subset with pairwise distinct slopes

Our results rely on probabilistic methods, results from incidence geometry, hypergraph containers, and additive combinatorics.

Part (II) is joint work with József Balogh, Felix Christian Clemen, and Dingyuan Liu.