Asymptotic Structure for the Clique Density Theorem

The famous Erdos-Rademacher problem asks for the smallest number of r -cliques in a graph with the given number of vertices and edges. Despite decades of active attempts, the asymptotic value of this extremal function for all r was determined only recently, by Reiher [Annals of Mathematics, 184 (2016) 683-707]. Here we describe the asymptotic structure of all almost extremal graphs. This task for r = 3 was previously accomplished by Pikhurko and Razborov [Combinatorics, Probability and Computing, 26 (2017) 138-160].