A conjecture of Erdős and Hajnal from 1989 says that forbidding any specific substructure results in existence of a very large homogeneous one! In this article you will have a look into one of the most fascinating problems in modern graph theory.
In a seminar talk in Cambridge this week, Julian Sahasrabudhe announced that he, together with his colleagues Marcelo Campos, Simon Griffiths and Rob Morris, had obtained an exponential improvement to the upper bound for Ramsey's theorem.
It is was the second time yesterday in a one week time and the fourth in a one month time that I came across Ramsey numbers. In the beginning I thought it was just a coincidence.