Large joints in graphs

TL;DR

This paper proves that large graphs with many r-cliques resemble the r-partite Turan graph structurally, unless they contain significantly more (r+1)-cliques sharing an edge, extending previous extremal graph theory results.

Contribution

It generalizes a known extremal graph theory result to larger classes of graphs, establishing a structural characterization involving (r+1)-cliques sharing edges.

Findings

Graphs with many r-cliques are structurally close to the Turan graph.

Such graphs contain more than Cn^(r-1) (r+1)-cliques sharing an edge unless they are Turan graphs.

The result extends previous extremal graph theory theorems.

Abstract

We show that if G is a graph of sufficiently large order n containing as many r-cliques as the r-partite Turan graph of order n; then for some C>0 G has more than Cn^(r-1) (r+1)-cliques sharing a common edge unless G is isomorphic to the the r-partite Turan graph of order n. This structural result generalizes a previous result that has been useful in extremal graph theory.