Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Ramsey-Turán problem for 4-cliques

Béla Csaba

TL;DR

The paper addresses the Ramsey-Turán problem for $K_4$ by deriving an essentially tight, Regularity-free bound on the maximum number of edges in $K_4$-free graphs with small independence number. It avoids the Regularity lemma by combining a minimum-degree reduction, a convexity-driven density analysis, and a new large, potentially unbalanced, quasi-random bipartite subgraph extraction, followed by an edge-reassignment step that preserves $K_4$-freeness while enabling sharp edge-count control. The main result shows that for suitably small $\alpha$ and large $n$, if $e(G) > \frac{n^2+n}{8} + (\alpha-\alpha^2)\frac{n^2}{2}$ then $G$ must contain a $K_4$, with constants that are single-exponential and Regularity-lemma-free. This advances Ramsey-Turán theory for $4$-cliques by delivering a practically tighter threshold over a broad range of $\alpha$ and improving the methodology beyond Regularity-based proofs.

Abstract

We present an essentially tight bound for the Ramsey-Turán problem for 4-cliques without using the Regularity lemma. This enables us to substantially extend the range in which one has the tight bound for the number of edges in $K_4$-free graphs as a function of the independence number, apart from lower order terms.

On the Ramsey-Turán problem for 4-cliques

TL;DR

The paper addresses the Ramsey-Turán problem for by deriving an essentially tight, Regularity-free bound on the maximum number of edges in -free graphs with small independence number. It avoids the Regularity lemma by combining a minimum-degree reduction, a convexity-driven density analysis, and a new large, potentially unbalanced, quasi-random bipartite subgraph extraction, followed by an edge-reassignment step that preserves -freeness while enabling sharp edge-count control. The main result shows that for suitably small and large , if then must contain a , with constants that are single-exponential and Regularity-lemma-free. This advances Ramsey-Turán theory for -cliques by delivering a practically tighter threshold over a broad range of and improving the methodology beyond Regularity-based proofs.

Abstract

We present an essentially tight bound for the Ramsey-Turán problem for 4-cliques without using the Regularity lemma. This enables us to substantially extend the range in which one has the tight bound for the number of edges in -free graphs as a function of the independence number, apart from lower order terms.

Paper Structure

This paper contains 5 sections, 6 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Definitions, main tools
  3. On the minimum degree of $G$
  4. Finding a large quasi-random pair
  5. Proof of the main theorem

Key Result

Theorem 1.1

For every $\eta>0$ there exists an $\alpha>0$ such that the following holds. Every $n$-vertex graph having at least $(\frac{1}{8}+\eta) n^2$ edges contains either a $K_4$ or an independent set larger than $\alpha n.$

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Claim 2.1
  • Claim 2.2
  • Definition 2.3
  • Claim 2.4
  • Lemma 2.5
  • Claim 2.6
  • Claim 3.3
  • ...and 11 more