The critical window for the classical Ramsey-Tur\'an problem

TL;DR

This paper investigates the critical edge density window in the classical Ramsey-Turán problem, providing nearly optimal bounds and resolving longstanding open questions about the independence number in this regime.

Contribution

It offers nearly best-possible bounds for the independence number in the critical edge density window, addressing key open problems in the classical Ramsey-Turán problem.

Findings

Established bounds close to the conjectured optimal in the critical window.

Solved several longstanding open problems in the Ramsey-Turán theory.

Enhanced understanding of the structure of K_4-free graphs near the extremal edge density.

Abstract

The first application of Szemer\'edi's powerful regularity method was the following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.