Joint Mathematics Colloquium ICMAT-UAM-UCM-UC3M

Analytic approach to
extremal combinatorics

Daniel Kráľ
(Masaryk University)

23 June 2023 - 12:00
Aula 520, Módulo 17, Dpto. de Matemáticas, UAM

Streaming: youtube.com/live/-r-J7vD4BEk

https://www.youtube.com/watch?v=-r-J7vD4BEk&ab_channel=Matem%C3%A1ticasUAM

 Analytic tools to represent and study large discrete structures provided by
the theory of combinatorial limits led to new views on a wide range of topics
in mathematics and computer science. After introducing the theory of
combinatorial limits, we will apply its methods to several specific problems
from extremal combinatorics and particularly from Ramsey theory. Ramsey
theory statements guarantee the existence of ordered substructures in large
objects such as in the following classical statement proven by Ramsey in
1930: if N is sufficiently large, then for any partition of k-tuples of N points
into finitely many classes, there exist n points such that all k-tuples formed
by these n points belong to the same class. We will study quantitative
versions of Ramsey type statements and present a solution of a 30-year-old
problem on the existence of high chromatic graphs with small Ramsey
multiplicity. In relation to general questions concerning the interplay of
combinatorial limits and extremal combinatorics, we will present, among
others, a counterexample to a conjecture of Lovász on finitely forcible
optima of extremal combinatorics problems.