The clique density theorem

Abstract

Abstract. Turán's theorem is a cornerstone of extremal graph theory. It asserts that for any integer r ě 2 every graph on n vertices with more than r´2 2pr´1q¨n 2 edges contains a clique of size r, i.e., r mutually adjacent vertices. The corresponding extremal graphs are balanced pr´1q-partite graphs.The question as to how many such r-cliques appear at least in any n-vertex graph with γn 2 edges has been intensively studied in the literature. In particular, Lovász andSimonovits conjectured in the 1970s that asymptotically the best possible lower bound is given by the complete multipartite graph with γn 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. §1. Introduction

On the Number of 4-Cycles in a Tournament

“…The recent works of Razborov [7], Nikiforov [9], and Reiher [10] solved the problem for s = 2 (with r = 3, r = 4, and all r ≥ 5, respectively). For all we know the problem is still open for s = 3 and any r.…”

Graphs with Few 3‐Cliques and 3‐Anticliques are 3‐Universal

“…Razborov's recent solution for was a major achievement in local graph theory. The problem was subsequently solved for by Nikiforov , and for , and general r by Reiher . To the best of our knowledge, the problem remains open for . Proof of Proposition Inequality : Recall that and .…”

“…At this writing even 3 is not yet fully understood (but see [11,16]). The state of our knowledge of l for l ≥ 4 is really very limited, though some work already exists, e.g., [7,8,12,13,[17][18][19]. Much of the recent progress in this area was achieved using Razborov's flag algebras method.…”

Rainbow triangles and the Caccetta‐Häggkvist conjecture

“…Goodman's theorem was improved by Bollobás [1], and more recently Razborov [18] proved a theorem establishing the precise dependence between the edge density and triangle density. (This result was then extended by Nikiforov [16] to K 4 subgraphs and by Reiher [20] for cliques of arbitrary size.) However, for the purpose of our application, using Razborov's theorem instead of Goodman's provides no improvement.…”