This video talks about a branch of Combinatorics, namely Design Theory. We look over a famous problem, definitions, visualisations and some theorems.
Notes:
I said that the Erdös-Hanani Conjecture was proved by Rödl in 1850. That is wrong. It was proved in 1985, 1850 being the year in which Kirkman's Schoolgirl Problem was published.
Time stamps:
0:00 - Intro
0:32 - Content summary
1:16 - Kirkman’s Schoolgirl Problem
2:06 - Visualisation
3:11 - Steiner Systems and Designs
6:12 - Hypergraphs
8:00 - From designs to Graphs
9:01 - Erdös-Hanani Conjecture
10:59 - Counting designs and conclusion
Quantamagazine article:
www.quantamagazine.org/mathematicians-find-hidden-structure-in-a-common-type-of-space-20230412/
Peter Keevash's articles on the existence of designs:
arxiv.org/abs/1401.3665
arxiv.org/abs/1802.05900
people.maths.ox.ac.uk/keevash/ - all articles written by Peter Keevash