# Asymptotic Structure for the Clique Density Theorem

## TL;DR

This paper characterizes the asymptotic structure of graphs that nearly minimize the number of r-cliques for given vertices and edges, building on recent solutions to the Erdős-Rademacher problem.

## Contribution

It extends the understanding of extremal graph structures for all r, providing a comprehensive description of almost extremal graphs in the clique density problem.

## Key findings

- Describes the asymptotic structure of almost extremal graphs for all r.
- Builds on Reiher's recent solution to the extremal function.
- Complements previous work on the case r=3.

## Abstract

The famous Erd\H{o}s-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].