Abstract

An efficient container lemma

Highlights

  • The hypergraph container theorems, proved several years ago by Balogh, Morris, and Samotij [6] and, independently, by Saxton and Thomason [42], state that the family of independent sets of any uniform hypergraph whose edges are distributed somewhat evenly may be covered by a small collection of sets, called containers, each of which is nearly independent

  • The two major reasons for this are the general form of the theorems and the explicit, optimal dependence between the various parameters disguised under the vague phrases ‘small collection’, ‘evenly distributed’, and ‘nearly independent’ above

  • In these applications of the container method to questions in Ramsey theory [12, 40], discrete geometry [9], and extremal graph theory [5, 33], the explicit dependence between uniformity and other parameters turned out to lie at the heart of the matter, obstructing the way to obtaining optimal bounds for several well studied functions

Read more

Summary

Introduction

The hypergraph container theorems, proved several years ago by Balogh, Morris, and Samotij [6] and, independently, by Saxton and Thomason [42], state that the family of independent sets of any uniform hypergraph whose edges are distributed somewhat evenly may be covered by (the families of subsets of) a small collection of sets, called containers, each of which is nearly independent. The container theorems have been used to analyse sequences of hypergraphs whose uniformities grow with the numbers of vertices and edges In these applications of the container method to questions in Ramsey theory [12, 40], discrete geometry [9], and extremal graph theory [5, 33], the explicit dependence between uniformity and other parameters turned out to lie at the heart of the matter, obstructing the way to obtaining optimal bounds for several well studied functions. In order to demonstrate this, we shall present four applications of our new theorem to problems in extremal graph theory, discrete geometry, and Ramsey theory that had previously been attacked using the original container theorems, obtaining an essential improvement of the state-of-the-art result in each case. We just point out here that one of these applications, an efficient version of the classical theorem of Kolaitis, Prömel, and Rothschild [28], Theorem 1.2 below, strongly suggests that Theorem 1.1 is, up to lower-order terms, optimal for general hypergraphs of large uniformities (see the discussion below the statement of Theorem 1.2)

The typical structure of graphs with no large cliques
Lower bounds on ε-nets
Upper bounds on Ramsey numbers
Packaged statement
Organisation of the paper
A word of motivation
Degree measures
The main technical result
The simple and packaged versions
Norms of degree measures
Degree measures and link hypergraphs
Linear combinations of degree measures
Outline
The key lemma
Pruning hypergraphs
The algorithm
Basic properties of the algorithm and the key dichotomy
The geometric lemma
Proof of the key dichotomy property
Proof of the key lemma
Probabilistic inequalities
An efficient container lemma for Kr+1-free graphs
Almost all Kr+1-free graphs are almost r-partite
Balanced and unbalanced r-partitions
The number of Kr+1-free graphs with a monochromatic star
The number of Kr+1-free graphs with a monochromatic matching
Findings
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call