Turan's theorem inverted

TL;DR

This paper explores the implications of Turan's theorem, demonstrating that graphs exceeding Turan's edge count contain significantly larger complete subgraphs and establishing stability results, thus extending Erdős's 1963 work.

Contribution

It provides new insights into the structure of graphs with many edges, showing larger complete subgraphs and stability properties, expanding classical Turan's theorem.

Findings

Graphs with more edges than Turan's threshold contain larger complete subgraphs.

Established stability theorems related to Turan's configurations.

Extended Erdős's 1963 work on extremal graph theory.

Abstract

Turan's theorem implies that every graph of order n with more edges than the r-partite Turan graph contains a complete graph of order r+1. We show that the same premise implies the existence of much larger graphs. We also prove corresponding stability theorems. These results complete work started by Erdos in 1963.