Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

K_4-free graphs have sparse halves

Christian Reiher

Abstract

Every $K_4$-free graph on $n$ vertices has a set of $\lfloor n/2\rfloor$ vertices spanning at most $n^2/18$ edges.

K_4-free graphs have sparse halves

Abstract

Every -free graph on vertices has a set of vertices spanning at most edges.

Paper Structure

This paper contains 4 sections, 4 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parity
  4. Ramsey-Turán theory

Key Result

Theorem 1.1

If a graph $G$ on $n$ vertices has the property that every set $X\subseteq V(G)$ of size $|X|=\lfloor \frac{1}{2}n\rfloor$ spans at least $\frac{1}{18}n^2$ edges, then either $G$ contains a $K_4$ or $n$ is divisible by $6$ and $G$ is a tripartite Turán graph.

Theorems & Definitions (5)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3: Ł uczak, Polcyn, and Reiher
  • proof : Proof of Lemma \ref{['lem:1243']}