Colorful Extensions of Infinite (p,q)-Theorems in Combinatorial Geometry

Speaker:
Organiser:
Arkadev Chattopadhyay
Date:
Tuesday, 27 Aug 2024, 16:00 to 17:00
Venue:
A-201 (STCS Seminar Room)
Category:
Abstract
An infinite sequence of sets \{B_n\}_{n\in\mathbb{N}} is said to be a heterochromatic sequence for an infinite collection \left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}} of families of sets, if there exists a strictly increasing sequence of natural numbers \left\{ i_{n}\right\}_{n \in \mathbb{N}} such that for all n \in \mathbb{N} we have B_{n} \in \mathcal{F}_{i_{n}}. In this talk, we will prove  that if for each n\in\mathbb{N}, \mathcal{F}_n is a family of nicely shaped convex sets in \mathbb{R}^d such that each heterochromatic sequence \{B_n\}_{n\in\mathbb{N}} of \left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}} contains k+2 sets that can be pierced by a single k -flat ( k-dimensional affine space) then all but finitely many \mathcal{F}_n's can be pierced by finitely many k-flats. This result generalizes the (\aleph_0, k+2)-Theorem proved by  Keller and Perles (SoCG'22) to the countably colorful setting. We have also established the tightness of our results by proving several no-go theorems.
This is a joint work with Sutanoya Chakraborty (PhD Student at ISI, Kolkata) and Soumi Nandi (PhD Student at ISI, Kolkata).

Short Bio:

Arijit Ghosh is currently an Associate Professor at ACM Unit, Indian Statistical Institute, Kolkata. He did his PhD in Computer Science from INRIA, France, and was a Postdoc at Max Planck Insitute for Informatics, Germany.