Simon's conjecture is a long-standing open problem in Combinatorics. In the first part of the seminar, we provide its motivation and all the necessary definitions to formulate it. In the second part, we present some recent improvements on this topic, pointing out the main tools used to obtain them and the typical approach to deal with this kind of problems.

Simon's conjecture is a long-standing open problem in Combinatorics. In the first part of the seminar, we provide its motivation and all the necessary definitions to formulate it. In the second part, we present some recent improvements on this topic, pointing out the main tools used to obtain them and the typical approach to deal with this kind of problems.