The Erdos-Gallai theorem states that every graph of average degree $d$ contains a cycle of length at least $d$. Alp Muyesser, Yuval Wigderson and I recently proved a robust and a hyperstability version of this theorem. At the core of the argument lies a very general structure theorem about graphs that originates from results of Pokrovskiy concerning the hyperstability of bounded-degree trees. 

In this talk, I will focus on explaining and motivating this structure theorem as well as providing some intuition on how to apply it.