The Clique Density Theorem

TL;DR

This paper proves a longstanding conjecture in extremal graph theory, establishing the minimum number of r-cliques in graphs with a given number of edges, generalizing previous results for specific cases.

Contribution

It confirms the Lovász-Simonovits conjecture for all r, determining the minimal number of r-cliques in graphs with a fixed edge density.

Findings

Proves the Lovász-Simonovits conjecture for all r.

Identifies the extremal graphs minimizing r-cliques.

Extends previous results from r=3,4 to all r.

Abstract

Tur\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geq 2$ every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually adjacent vertices. The corresponding extremal graphs are balanced $(r-1)$-partite graphs. The question as to how many such $r$-cliques appear at least in any $n$-vertex graph with $\gamma n^2$ edges has been intensively studied in the literature. In particular, Lov\'{a}sz and Simonovits conjectured in the 1970s that asymptotically the best possible lower bound is given by the complete multipartite graph with $\gamma n^2$ edges in which all but one vertex class is of the same size while the remaining one may be smaller. Their conjecture was recently resolved for $r=3$ by Razborov and for $r=4$ by Nikiforov. In this article, we prove the conjecture for all values of $r$.