A Weighted Regularity Lemma with Applications

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Research Article

A Weighted Regularity Lemma with Applications

Béla Csaba

1

and András Pluhár

2

1_{Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6724, Hungary} 2_{Department of Computer Science, University of Szeged, ´}_{Arpád tér 2, Szeged 6720, Hungary}

Correspondence should be addressed to B´ela Csaba; [email protected]

Received 13 February 2014; Revised 27 May 2014; Accepted 27 May 2014; Published 19 June 2014 Academic Editor: Laszlo A. Szekely

Copyright © 2014 B. Csaba and A. Pluh´ar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove an extension of the regularity lemma with vertex and edge weights which in principle can be applied for arbitrary graphs. The applications involve random graphs and a weighted version of the Erd˝os-Stone theorem. We also provide means to handle the otherwise uncontrolled exceptional set.

1. Introduction

Let𝐺 = 𝐺(𝐴, 𝐵)be a bipartite graph. For𝑋, 𝑌 ⊂ 𝐴 ∪ 𝐵let

𝑒𝐺(𝑋, 𝑌)denote the number of edges with one endpoint in𝑋

and the other in𝑌. Given an𝜀 > 0we say that the(𝐴, 𝐵)-pair

is𝜀-regular if 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 𝑒𝐺(𝐴󸀠, 𝐵󸀠) 󵄨󵄨󵄨󵄨𝐴󸀠󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝐵󸀠󵄨󵄨󵄨󵄨 − 𝑒_𝐺(𝐴, 𝐵) |𝐴| |𝐵| 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨󵄨< 𝜀 (1)

for every𝐴 ⊂ 𝐴,|𝐴󸀠| > 𝜀|𝐴|and𝐵󸀠⊂ 𝐵,|𝐵󸀠| > 𝜀|𝐵|.

This definition plays a crucial role in the celebrated

Regularity Lemma of Szemer´edi; see [1, 2]. The regularity

lemma is a very powerful tool when applied to a dense graph. It has found lots of applications in several areas of mathemat-ics and computer science; for applications in graph theory

see for example, [3]. However, it does not tell us anything

useful when applied for a sparse graph (i.e., a graph on

𝑛vertices having𝑜(𝑛2)edges).

There has been significant interest to find widely appli-cable versions for sparse graphs. This turns out to be a very

hard task. Kohayakawa [4] proved a sparse regularity lemma,

and with Kohayakawa et al. [5] they applied it for finding

arithmetic progressions of length 3 in dense subsets of a random set. In their sparse regularity lemma dense graphs are substituted by dense subgraphs of a random (or

quasir-andom) graph. Naturally, a new definition of 𝜀-regularity

was needed; below we formulate a slightly different version from theirs.

Let𝐹(𝐴, 𝐵)and𝐺(𝐴, 𝐵)be two bipartite graphs such that

𝐹 ⊂ 𝐺. We say that the(𝐴, 𝐵)-pair is𝜀-regular in𝐹relative to

𝐺if 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 𝑒_𝐹(𝐴󸀠_{, 𝐵}󸀠₎ 𝑒_𝐺(𝐴󸀠_{, 𝐵}󸀠₎ − 𝑒_𝐹(𝐴, 𝐵) 𝑒𝐺(𝐴, 𝐵) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨< 𝜀 (2)

for every𝐴󸀠 ⊂ 𝐴,𝐵󸀠 ⊂ 𝐵and|𝐴󸀠| > 𝜀|𝐴|,|𝐵󸀠| > 𝜀|𝐵|. It is

easy to see that the above is a generalization of𝜀-regularity; in

the original definition the role of𝐺is played by the complete

bipartite graph𝐾_𝐴,𝐵. In this more general definition𝐹can be

a rather sparse graph; it only has to be denserelative to𝐺; that

is,𝑒(𝐹)/𝑒(𝐺)should be a constant.

In this paper we further generalize the notion of

quasiran-domness and𝜀-regularity by introducingweighted regularity

using vertex and edge weights. This enables us to prove a more general and perhaps more applicable regularity lemma. Let us remark that another notion of regularity is used by Alon

et al. [6]; later we will discuss how their work relates to ours.

A recent approach by Scott [7] defines regularity of matrices

and deduces a regularity lemma for graphs via their adjacency matrices. This approach turns out to be less flexible than the one we choose in the present paper (for earlier versions, see

[8]).

The basic tool is the Strong Structure Theorem of Tao [9],

where he simplifies the proof of the original regularity lemma

Volume 2014, Article ID 602657, 9 pages http://dx.doi.org/10.1155/2014/602657

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itself and gives new insights, too. Following his lines became technically feasible to extend regularity to the case when both the edges and the vertices of a graph are weighted (note that the measures are in close connection with each other.) We remark that similar ideas might be used to find a regularity lemma for sparse hypergraphs as well.

The structure of the paper is as follows. First we discuss

weighted quasirandomness and weighted𝜀-regularity in the

second section. In the third section we prove the new version of the regularity lemma. Finally, we show some applications in the fourth section; in particular, we prove a weighted version of the Erd˝os-Stone theorem.

2. Basic Definitions and Tools

Throughout the paper we apply the relation “≪”: 𝑎 ≪ 𝑏

if 𝑎 is sufficiently smaller than 𝑏. This notation is widely

applied in papers using the regularity lemma and simplifies our notation, too.

Let 𝛽 > 0 and 𝐺 = (𝑉, 𝐸) be a graph on 𝑛vertices. Set𝛿_𝐺 = 𝑒(𝐺)/ (𝑛₂); this is thedensity of𝐺. We define the

density of the𝐴, 𝐵pair of subsets of𝑉(𝐺) by𝛿_𝐺(𝐴, 𝐵) =

𝑒_𝐺(𝐴, 𝐵)/(|𝐴||𝐵|). We say that𝐺is𝛽-quasi-random if it has

the following property: If𝐴, 𝐵 ⊂ 𝑉(𝐺)such that𝐴 ∩ 𝐵 = 0

and|𝐴|, |𝐵| > 𝛽𝑛then

󵄨󵄨󵄨󵄨𝛿𝐺− 𝛿𝐺(𝐴, 𝐵)󵄨󵄨󵄨󵄨 < 𝛽𝛿𝐺. (3)

That is, the edges of𝐺are distributed “randomly.” In order

to formulate our regularity lemma we have to define quasir-andomness in a more general way that admits weights on vertices and edges.

For a function𝑤 : 𝑆 → R+and𝐴 ⊂ 𝑆,𝑤(𝐴)is defined

by the usual way; that is,𝑤(𝐴) = ∑_𝑥∈𝐴𝑤(𝑥). We will also

use the indicator function of the edge set of a graph𝐻.1_𝐻:

(𝑉(𝐻)

2 ) → {0, 1}and1𝐻(𝑥, 𝑦) = 1if 𝑥𝑦 ∈ 𝐸(𝐻).

We define the weighted quasirandomness of a graph𝐺 =

(𝑉, 𝐸)with given weight-functions𝜇 : 𝑉 → R+ and𝜌 :

(𝑉

2) → R+. For𝐴, 𝐵 ⊂ 𝑉let

𝜌_𝐺(𝐴, 𝐵) := ∑

𝑢∈𝐴,V∈𝐵

1_𝐺(𝑢,V) 𝜌 (𝑢,V) . ₍₄₎

In particular,𝜌_𝐺(𝑢,V) =1_𝐺(𝑢,V)𝜌(𝑢,V)for𝑢,V∈ 𝑉. Observe

that the function 𝜇 is an analogon of the vertex counting

function on a set, while the function𝜌counts the edges in

the unweighted case.

Definition 1. A graph𝐺 = (𝑉, 𝐸)is weighted𝛽-quasi-random

with weight-function𝜇and𝜌if for any𝐴, 𝐵 ⊂ 𝑉(𝐺)such that

𝐴 ∩ 𝐵 = 0and𝜇(𝐴) ≥ 𝛽𝜇(𝑉),𝜇(𝐵) ≥ 𝛽𝜇(𝑉)one has

󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨𝜇 (𝐴) 𝜇 (𝐵)𝜌𝐺(𝐴, 𝐵) −

𝜌_𝐺(𝑉, 𝑉)

𝜇 (𝑉) 𝜇 (𝑉)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 < 𝛽. (5)

Observe that choosing𝜇 ≡ 1and𝜌 ≡ 1/𝛿_𝐺gives back the

first definition of quasirandomness. The notion of quasiran-domness readily extends to bipartite (or multipartite) graphs.

In that case the sets𝐴and𝐵are chosen from different classes.

There is another weaker notion of quasirandomness, which we will also use.

Definition 2. Let𝐾 > 1be an absolute constant. A graph

𝐺 = (𝑉, 𝐸)is weighted(𝐾, 𝛽)-quasi-random with

weight-functions𝜇and𝜌if for any𝐴, 𝐵 ⊂ 𝑉(𝐺)such that𝐴 ∩ 𝐵 = 0

and𝜇(𝐴) ≥ 𝛽𝜇(𝑉),𝜇(𝐵) ≥ 𝛽𝜇(𝑉)one has

1 𝐾 𝜌_𝐺(𝑉, 𝑉) 𝜇 (𝑉) 𝜇 (𝑉)≤ 𝜌_𝐺(𝐴, 𝐵) 𝜇 (𝐴) 𝜇 (𝐵) ≤ 𝐾 𝜌_𝐺(𝑉, 𝑉) 𝜇 (𝑉) 𝜇 (𝑉). (6)

Clearly, if a graph is𝛽-quasi-random and𝐾 > max{1 +

𝛽/𝑦, 1 + 𝛽/(𝑦 − 𝛽)}, then it is(𝐾, 𝛽)-quasi-random, where

𝑦 := 𝜌_𝐺(𝑉, 𝑉)/𝜇(𝑉)2. Now we need to describe the weighted version of relative regularity.

Definition 3. Let𝐺and𝐹be graphs,𝐹 ⊂ 𝐺, and assume that

𝐺is a(𝐾, 𝛽)-quasi-random with weight functions𝜇and 𝜌

as defined above. For𝐴, 𝐵 ⊂ 𝑉(𝐺)and𝐴 ∩ 𝐵 = 0the pair

(𝐴, 𝐵)in𝐹is(𝜇, 𝜌)-weighted𝜖-regular relative to𝐺, or briefly

weighted𝜖-regular, if 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 𝜌𝐹(𝐴󸀠, 𝐵󸀠) 𝜇 (𝐴󸀠_{) 𝜇 (𝐵}󸀠₎− 𝜌_𝐹(𝐴, 𝐵) 𝜇 (𝐴) 𝜇 (𝐵) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨< 𝜖 (7)

for every𝐴󸀠 ⊂ 𝐴and𝐵󸀠 ⊂ 𝐵provided that𝜇(𝐴󸀠) ≥ 𝜖𝜇(𝐴),

𝜇(𝐵󸀠_{) ≥ 𝜖𝜇(𝐵)}_{. Here}

𝜌_𝐹(𝐴, 𝐵) = ∑

1_𝐹(𝑢,V) 𝜌 (𝑢,V) . ₍₈₎

Remarks. Note that weighted𝜖-regularity is nothing but the

well-known𝜖-regularity when𝐺 = 𝐾_𝐴,𝐵 and𝜇 ≡ 1and𝜌

is chosen to be identically the reciprocal of the density of𝐺

as before. Since1_𝐹(𝑢,V) ≤ 1_𝐺(𝑢,V) ≤ 1(𝑢,V)we also have

𝜌_𝐹(𝐴, 𝐵) ≤ 𝜌_𝐺(𝐴, 𝐵) ≤ 𝜌(𝐴, 𝐵). Hence, the first inequality

of the definition does not refer to𝐺explicitly but contains

information on it.

Next we define weighted regular partitions.

Definition 4. Let𝐺 = (𝑉, 𝐸)and𝐹 ⊂ 𝐺be graphs, and𝜇and𝜌

weight functions.𝐹has a weighted𝜖-regular partition relative

to𝐺if its vertex set𝑉can be partitioned intoℓ + 1clusters

𝑊₀, 𝑊₁, . . . , 𝑊_ℓsuch that (i)𝜇(𝑊₀) ≤ 𝜖𝜇(𝑉),

(ii)|𝜇(𝑊_𝑖)−𝜇(𝑊_𝑗)| ≤max_𝑥∈𝑉{𝜇(𝑥)}for every1 ≤ 𝑖,𝑗 ≤ ℓ,

(iii) all but at most𝜖ℓ2of the pairs(𝑊_𝑖, 𝑊_𝑗)for1 ≤ 𝑖 < 𝑗 ≤

ℓare weighted𝜖-regular in𝐹relative to𝐺.

In order to show our main result we will use the Strong Structure Theorem of Tao that allows a short exposition. In fact we will closely follow his proof for the regularity lemma

as discussed in [9].

First we have to introduce some definitions. Let𝐻be a

real finite-dimensional Hilbert space, and let𝑆be a set of basic

functions or basic structured vectors of𝐻of norm at most 1.

The function𝑔 ∈ 𝐻is(𝑀, 𝐾)-structured with the positive

integers𝑀,𝐾if one has a decomposition

𝑔 = ∑

1≤𝑖≤𝑀

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with𝑠_𝑖 ∈ 𝑆and𝑐_𝑖 ∈ [−𝐾, 𝐾]for1 ≤ 𝑖 ≤ 𝑀. We say that𝑔is

𝛽-pseudorandom for some𝛽 > 0if|⟨𝑔, 𝑠⟩| ≤ 𝛽for all𝑠 ∈ 𝑆.

Then we have the following.

Theorem 5 (Strong Structure Theorem—T. Tao). Let𝐻and

𝑆 be as above, let 𝜀 > 0, and let 𝐽 : Z+ → R+ be an arbitrary function. Let𝑓 ∈ 𝐻be such that‖𝑓‖_𝐻 ≤ 1. Then we can find an integer𝑀 = 𝑀_𝐽,𝜀and a decomposition𝑓 =

𝑓str+ 𝑓psd+ 𝑓errwhere (i)𝑓stris(𝑀, 𝑀)-structured, (ii)𝑓psdis

1/𝐽(𝑀)-pseudorandom, and (iii)‖𝑓err‖𝐻≤ 𝜀.

Note that the proof of Theorem 5 yields a polynomial

algorithm; hence, our regularity lemma has the same com-plexity.

3. Weighted Regularity Lemma Relative to

a Quasirandom Graph

𝐺

First we define the Hilbert space 𝐻, and 𝑆. We generalize

Example 2.3 of [9] to weighted graphs. Let𝐺 = (𝑉, 𝐸)be a

𝛽-quasi-random graph on𝑛vertices with weight functions

𝜇and 𝜌. Let𝐻be the(𝑛₂)-dimensional space of functions

𝑔 : (𝑉

2) → R, endowed with the inner product

⟨𝑔, ℎ⟩ = 1 (𝑛

2)_(𝑢,_V∑ )∈( 𝑉2)

𝑔 (𝑢,V) ℎ (𝑢,V) 𝜌𝐺(𝑢,V) . ₍₁₀₎

It is useful to normalize the vertex and edge weight

functions; we assume that𝜇(𝑉) = 𝑛and⟨1, 1⟩ = 1. We also

assume that𝜇(V) = 𝑜(|𝑉|)for everyV ∈ 𝑉. Observe that if

𝐹 ⊂ 𝐺then‖1_𝐹‖ ≤ 1. We let𝑆be the collection of 0,1-valued

functions𝛾_𝐴,𝐵for𝐴, 𝐵 ⊂ 𝑉(𝐺),𝐴∩𝐵 = 0, where𝛾_𝐴,𝐵(𝑢,V) = 1

if and only if𝑢 ∈ 𝐴andV∈ 𝐵. We have the following.

Theorem 6 (weighted regularity lemma). Let 𝐾 > 1 and

𝛽, 𝜀 ∈ (0, 1), such that0 < 𝛽 ≪ 𝜀 ≪ 1/𝐾, and let𝐿 ≥ 1. If𝐺 = (𝑉, 𝐸)is a weighted(𝐾, 𝛽)-quasi-random graph on𝑛 vertices with𝑛sufficiently large depending on𝜀and𝐿,𝐹 ⊂ 𝐺, then𝐹admits a weighted𝜀-regular partition relative to𝐺into the partition sets𝑊₀, 𝑊₁, . . . , 𝑊_ℓsuch that𝐿 ≤ ℓ ≤ 𝐶_𝜀,𝐿for some constant𝐶_𝜀,𝐿.

Proof. Let us apply Theorem 5 to the function 1_𝐹 with

parameters𝜂and function𝐽to be chosen later. We get the

decomposition

1_𝐹= 𝑓str+ 𝑓psd+ 𝑓err, (11)

where 𝑓str is (𝑀, 𝑀)-structured, 𝑓psd is 1/𝐽(𝑀)

-pseudorandom, and‖𝑓_err‖ ≤ 𝜂with𝑀 = 𝑀_𝐽,𝜂= 𝑀_𝐽,𝜀.

The function𝑓stris the combination of at most𝑀basic

functions:

𝑓str= ∑

1≤𝑘≤𝑀

𝛼_𝑘𝛾_A𝑘_,_B𝑘, ₍₁₂₎

whereA_𝑘,B_𝑘 are subsets of𝑉and 𝛾_A_𝑘_,_B_𝑘 agrees with the

indicator function of the edges of 𝐺 in between A_𝑘 and

B_𝑘. Any(A_𝑘,B_𝑘)pair partitions𝑉into at most 4 subsets.

Overall we get a partitioning of𝑉into at most4𝑀subsets; we

will refer to them asatoms. Divide every atom into subsets of

total vertex weight𝜀𝑛/(𝐿 + 4𝑀), except possibly one smaller

subset. The small subsets will be put into𝑊₀; the others give

𝑊₁, 𝑊₂, . . . , 𝑊_ℓ, withℓ = (𝐿 + 4𝑀)/𝜀. We refer to the sets𝑊_𝑖 for𝑖 = 1, . . . , ℓasclusters. If𝑛is sufficiently large then this partitioning is nontrivial. From the construction it follows

that each𝑊_𝑖is entirely contained within an atom. It is also

clear that𝜇(𝑊₀) ≤ 𝜀𝑛and𝜇(𝑊_𝑖) ≈ 𝑚 = Θ(𝑛/ℓ)for every

1 ≤ 𝑖 ≤ ℓ. We have that 󵄩󵄩󵄩󵄩𝑓err󵄩󵄩󵄩󵄩2= ₍1𝑛 2)_(𝑢,_V∑ )∈( 𝑉2)󵄨󵄨󵄨󵄨𝑓 err(𝑢,V)󵄨󵄨󵄨󵄨2𝜌𝐺(𝑢,V) ≤ 𝜂2. ₍₁₃₎

From this and the normalization of𝜌it follows that

1 (ℓ 2)1≤𝑖<𝑗≤ℓ∑ 1 𝜌_𝐺(𝑊_𝑖, 𝑊_𝑗)𝑢∈𝑊∑𝑖,V∈𝑊𝑗 󵄨󵄨󵄨󵄨𝑓err(𝑢,V)󵄨󵄨󵄨󵄨2𝜌𝐺(𝑢,V) = 𝑂 (𝜂2) . (14) Clearly, 1 𝜌_𝐺(𝑊_𝑖, 𝑊_𝑗)𝑢∈𝑊∑𝑖,V∈𝑊𝑗 󵄨󵄨󵄨󵄨𝑓err(𝑢,V)󵄨󵄨󵄨󵄨2𝜌𝐺(𝑢,V) = 𝑂 (𝜂) (15)

for all but at most 𝑂(𝜂ℓ2) pairs (𝑖, 𝑗). If the above is

satisfied for a pair(𝑖, 𝑗)then we call it agood pair. We will

apply the Cauchy-Schwarz inequality. For that let𝑎(𝑢,V) =

|𝑓err(𝑢,V)|√𝜌𝐺(𝑢,V)and𝑏(𝑢,V) = √𝜌𝐺(𝑢,V); then

∑_𝑢∈𝑊_𝑖_,_V_∈𝑊_𝑗𝑎 (𝑢,V) 𝑏 (𝑢,V) √∑𝑢∈𝑊𝑖,V∈𝑊𝑗𝑏2(𝑢,V) ≤ √ ∑ 𝑢∈𝑊𝑖,V∈𝑊𝑗 𝑎2_(𝑢,_V_). ₍₁₆₎ Since √ ∑ 𝑢∈𝑊𝑖,V∈𝑊𝑗 𝑎2_(𝑢,_V_{) = 𝑂 (√𝜂) √𝜌} 𝐺(𝑊𝑖, 𝑊𝑗), (17) we get that 1 𝜌_𝐺(𝑊_𝑖, 𝑊_𝑗)𝑢∈𝑊∑𝑖,V∈𝑊𝑗 󵄨󵄨󵄨󵄨𝑓err(𝑢,V)󵄨󵄨󵄨󵄨 𝜌𝐺(𝑢,V) = 𝑂 (√𝜂) (18) if(𝑖, 𝑗)is a good pair.

Assume that (𝑖, 𝑗)is a good pair. From the

pseudoran-domness of𝑓psdwe have that

󵄨󵄨󵄨󵄨 󵄨⟨𝑓psd, 𝛾𝐴,𝐵⟩󵄨󵄨󵄨󵄨󵄨 = ₍1𝑛 2) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨𝑢∈𝐴,∑V∈𝐵 𝑓psd(𝑢,V) 𝜌𝐺(𝑢,V)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨≤ 1 𝐽 (𝑀) (19)

for every𝐴 ⊂ 𝑊_𝑖and𝐵 ⊂ 𝑊_𝑗.

We will show that every good pair is weighted𝜀-regular

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𝐴 ⊂ 𝑊_𝑖,𝜇(𝐴) > 𝜀𝜇(𝑊_𝑖)and𝐵 ⊂ 𝑊_𝑗,𝜇(𝐵) > 𝜀𝜇(𝑊_𝑗). To show

that(𝑊_𝑖, 𝑊_𝑗)is weighted𝜀-regular, it is sufficient to show that

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 𝜌_𝐹(𝐴, 𝐵) 𝜇 (𝐴) 𝜇 (𝐵)− 𝜌_𝐹(𝑊_𝑖, 𝑊_𝑗) 𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨< 𝜀. (20) Recall that 𝜌_𝐹(𝐴, 𝐵) = ∑ 𝑢∈𝐴,V∈𝐵 1_𝐹(𝑢,V) 𝜌 (𝑢,V) = ∑ 𝑢∈𝐴,V∈𝐵 1_𝐹(𝑢,V) 𝜌_𝐺(𝑢,V) , (21) since𝐹 ⊂ 𝐺.

Substituting𝑓str+ 𝑓psd+ 𝑓errfor1𝐹it is sufficient to verify

the following inequalities:

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 ∑_𝑢∈𝐴,_V_∈𝐵𝑓str(𝑢,V) 𝜌𝐺(𝑢,V) 𝜇 (𝐴) 𝜇 (𝐵) − ∑_𝑢∈𝑊_𝑖_,_V_∈𝑊_𝑗𝑓str(𝑢,V) 𝜌𝐺(𝑢,V) 𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 < 𝜀/3, (22) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 ∑_𝑢∈𝐴,_V_∈𝐵𝑓psd(𝑢,V) 𝜌𝐺(𝑢,V) 𝜇 (𝐴) 𝜇 (𝐵) − ∑_𝑢∈𝑊_𝑖_,_V_∈𝑊_𝑗𝑓psd(𝑢,V) 𝜌𝐺(𝑢,V) 𝜇 (𝑊_𝑖) 𝜇 (𝑊_𝑗) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 < 𝜀/3, (23) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 ∑_𝑢∈𝐴,_V_∈𝐵𝑓err(𝑢,V) 𝜌𝐺(𝑢,V) 𝜇 (𝐴) 𝜇 (𝐵) − ∑_𝑢∈𝑊_𝑖_,_V_∈𝑊_𝑗𝑓err(𝑢,V) 𝜌𝐺(𝑢,V) 𝜇 (𝑊_𝑖) 𝜇 (𝑊_𝑗) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 < 𝜀/3. (24)

For proving (22) recall that𝑓stris constant on𝑊𝑖×𝑊𝑗and

(𝑀, 𝑀)-structured. Since the𝛾_𝑋,𝑌 basic functions are0, 1

-valued, we get that|𝑓str| ≤ 𝑀2. Moreover,𝐺is(𝐾, 𝛽)

-quasi-random, where0 < 𝛽 ≪ 𝜀. Therefore, (22) follows from the

inequality𝐾𝑀2𝛽 < 𝜀/3, since𝛽 ≪ 𝜖.

The proof of (23) goes as follows. The first term is

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨 ∑_𝑢∈𝐴,_V_∈𝐵𝑓psd(𝑢,V) 𝜌𝐺(𝑢,V) 𝜇 (𝐴) 𝜇 (𝐵) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨= (𝑛2)󵄨󵄨󵄨󵄨󵄨⟨𝑓psd, 𝛾𝐴,𝐵⟩󵄨󵄨󵄨󵄨󵄨 ≤ (𝑛2) 𝐽 (𝑀) 𝜇 (𝐴) 𝜇 (𝐵), (25)

and the second is

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 ∑_𝑢∈𝑊_𝑖_,_V_∈𝑊_𝑗𝑓psd(𝑢,V) 𝜌𝐺(𝑢,V) 𝜇 (𝑊_𝑖) 𝜇 (𝑊_𝑗) 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨= (𝑛2)󵄨󵄨󵄨󵄨󵄨󵄨⟨𝑓psd, 𝛾𝑊𝑖,𝑊𝑗⟩󵄨󵄨󵄨󵄨󵄨󵄨 ≤ (𝑛2) 𝐽 (𝑀) 𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗) . (26)

Noting that𝜇(𝑊_𝑘) = Θ(𝑛/ℓ)for𝑘 ≥ 1we get that the sum of

the above terms is at most

ℓ2 2𝐽 (𝑀)(1 + 1 𝜀2) < 𝜀 3, (27) if𝐽(𝑀) ≫ ℓ2/𝜀3.

For (24) first notice that it is upper bounded by

𝑂 (√𝜂) ( 𝜌𝐺(𝑊𝑖, 𝑊𝑗) 𝜇 (𝑊_𝑖) 𝜇 (𝑊_𝑗)+ 𝜌_𝐺(𝑊_𝑖, 𝑊_𝑗) 𝜇 (𝐴) 𝜇 (𝐵)) ≤ 𝑂 (√𝜂) 𝜌𝐺(𝑊𝑖, 𝑊𝑗) 𝜀2_{𝜇 (𝑊} 𝑖) 𝜇 (𝑊𝑗) . (28)

We also have that

𝜌_𝐺(𝑊_𝑖, 𝑊_𝑗)

𝜇 (𝑊_𝑖) 𝜇 (𝑊_𝑗) = 𝑂 (1) (29)

by the normalization of𝜇and𝜌and from the fact that𝐺is

quasirandom. From this it is easy to see that if𝜂 ≪ 𝜀6then

(24) is at most𝜀/3. This finishes the proof of the theorem.

4. Quasirandom Weightings and Applications

In this section we first prove that a random graph with widely differing edge probabilities is quasirandom, if none of the edge probabilities are too small. In this case the vertex weights will all be one, but edges will have different weights. Then we show examples where vertices have different weights. We will consider the relation of weighted regularity and volume

regularity. We define the “natural weighting” of 𝐾_𝑛 and

prove a weighted version of the Erd˝os-Stone theorem for this weighting. Finally, we show how to partially control the non-exceptional set by natural weightings.

4.1. Quasirandomness in the𝐺(𝑛,𝑝_𝑖𝑗)Model. In this section

we will prove that random graphs of the𝐺(𝑛, 𝑝_𝑖𝑗)model are

quasirandom in the strong sense with high probability. A

special case of this model is the well-known𝐺(𝑛, 𝑝)model

for random graphs. A regularity lemma for this case was first

applied by Kohayakawa et al. [5]. They studied𝐺(𝑛, 𝑝)for

𝑝 = 𝑐/√𝑛in order to find arithmetic progressions of length

three in dense subsets of random subsets of[𝑁].

The 𝐺(𝑛, 𝑝_𝑖𝑗) model was first considered by Bollob´as

[10]. Recently it was also studied by Chung and Lu [11]. In

this model one takes𝑛vertices and draws an edge between

the vertices 𝑥_𝑖 and 𝑥_𝑗 with probability 𝑝_𝑖𝑗, randomly and

independently of each other. Note that if𝑝_𝑖𝑗 ≡ 𝑝, then we

get back the well-known𝐺(𝑛, 𝑝)model. It is a straightforward

application of the Chernoff bound that a random graph𝐺 ∈

𝐺(𝑛, 𝑝)is quasirandom with high probability if𝑝 ≫ 1/𝑛.

However, the case of𝐺(𝑛, 𝑝_𝑖,𝑗)is somewhat harder.

Lemma 7. Let𝛽 > 0. There exists a𝐾 = 𝐾(𝛽)such that if𝐺 ∈

𝐺(𝑛, 𝑝_𝑖𝑗)and𝑝_𝑖𝑗≥ 𝐾/𝑛for every𝑖and𝑗, then𝐺is weighted𝛽 -quasi-random with probability at least1−2−𝑛if𝑛is sufficiently large.

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Proof. First of all let𝜇 ≡ 1, and let𝜌(𝑖, 𝑗) = 1/𝑝_𝑖,𝑗. Set𝐾 =

4800/𝛽6_{. Let}_𝑝

0 = 𝐾/𝑛, and let𝑝𝑘 = 𝑒𝑘𝑝0for1 ≤ 𝑘 ≤log𝑛.

Let𝐴and 𝐵be a pair of disjoint sets, both of size at least

𝛽𝑛. We partition the pairs(𝑢,V), where𝑢 ∈ 𝐴andV ∈ 𝐵,

into𝑂(log𝑛)disjoint sets𝐻₁, 𝐻₂, . . . , 𝐻_𝑙: if𝑝_𝑘 ≤ 𝑝_𝑢_V < 𝑝_𝑘+1

then(𝑢,V)will belong to𝐻_𝑘. Let𝑎_𝑘= (𝛽3/10)√𝑒𝑘𝐾𝑛. We will

denote|𝐻_𝑘|by𝑚_𝑘.

We will prove that the following inequality holds with

probability at least1 − 2−3𝑛: 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨𝑢∈𝐴,∑V∈𝐵 𝑋_𝑢_V 𝑝_𝑢_V|𝐴| |𝐵|− 1󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨< 𝛽/2, (30)

where 𝑋_𝑢_V is a random variable which is 1 if 𝑢V ∈ 𝐸(𝐺);

otherwise it is 0. This implies the quasirandomness of 𝐺

since there are less than22𝑛pairs of disjoint subsets of𝑉(𝐺).

Observe that

∑

E𝑋_𝑢_V

𝑝_𝑢_V|𝐴| |𝐵| = 1. (31)

Applying the large deviation inequalities 𝐴.1.11 and

𝐴.1.13from [12], we are able to bound the number of edges in

between𝐴and𝐵for the edges of𝐻_𝑘in case𝑚_𝑘is sufficiently

large as follows. According to𝐴.1.11we have that

Pr( ∑ (𝑢,V)∈𝐻𝑘 (𝑋_𝑢_V−E𝑋_𝑢_V) > 𝑎_𝑘) < 𝑒−(𝑎2 𝑘/2𝑞𝑘𝑚𝑘)+(𝑎𝑘3/2𝑞2𝑘𝑚2𝑘), (32) where 𝑝_𝑘≤ 𝑞_𝑘 = ∑ (𝑢,V)∈𝐻𝑘 𝑝_𝑢_V 𝑚_𝑘 < 𝑝𝑘+1. (33)

We estimate the exponent in case𝑚_𝑘= 𝑛2:

− 𝑎𝑘2 2𝑞_𝑘𝑚_𝑘 + 𝑎3_𝑘 2𝑞2 𝑘𝑚2𝑘 ≤ −𝛽6 200( √𝑒2 𝑒 ) 𝑘 𝐾𝑛3 𝑒𝑚_𝑘 + 𝛽9 2000( √𝑒3 𝑒2 ) 𝑘 𝑒𝐾𝑛5 𝑚2 𝑘 < −3𝑛, (34)

where we used the definition of𝐾. For𝑚_𝑘 being much less

than𝑛2, direct substitution gives a useless bound. For this case

we have the useful inequality

1 2Pr( 𝑚𝑘 ∑ 𝑖=1𝑌𝑖> 𝑎𝑘) ≤ Pr( 𝑛2 ∑ 𝑖=1𝑌𝑖> 𝑎_𝑘 2) , (35)

where Pr(𝑌_𝑖 = 1 − 𝑞_𝑘) = 𝑞_𝑘and Pr(𝑌_𝑖 = −𝑞_𝑘) = 1 − 𝑞_𝑘. This

implies that the exponent is at most−3𝑛even in case𝑚_𝑘< 𝑛2.

Indeed, let𝐴,𝐵, and𝐶be the events that∑𝑚𝑘

𝑖=1𝑌𝑖 > 𝑎𝑘,

∑𝑛_𝑖=12 𝑌𝑖 > 𝑎𝑘/2, and∑𝑛

2

𝑖=𝑚𝑘+1𝑌𝑖 < −𝑎𝑘/2, respectively. Clearly

𝐴and𝐶are independent, and𝐴 ∩ 𝐶 ⊂ 𝐵. So we have Pr(𝐵) ≥

Pr(𝐴 ∩ 𝐶) = Pr(𝐴)Pr(𝐶); that is, Pr(𝐴) ≤ Pr(𝐵)/Pr(𝐶) < Pr(𝐵)/2, since by𝐴.1.13 Pr( 𝑛2 ∑ 𝑖=𝑚𝑘+1 𝑌_𝑖< −𝑎𝑘 2) < 𝑒−𝑎 2 𝑘/8𝑞𝑘(𝑛2−𝑚𝑘)< 1 2. (36)

With this we have proved that the sum of the weights of the

edges of𝐻_𝑘 will not be much larger than their expectation

with high probability.

Now we estimate the probability that the sum of the

weights is much less than their expectation. Let us use𝐴.1.13

again directly to the sums over𝐻_𝑘’s:

Pr( ∑

(𝑢,V)∈𝐻𝑘

(𝑋_𝑢_V−E𝑋_𝑢_V) < −𝑎_𝑘) < 𝑒−𝑎2

𝑘/2𝑞𝑘𝑚𝑘. ₍₃₇₎

The exponent in the inequality can be estimated very similarly as before: − 𝑎2𝑘 2𝑞_𝑘𝑚_𝑘 ≤ − 𝛽6 200( √𝑒2 𝑒 ) 𝑘 𝐾𝑛3 𝑒𝑚_𝑘 < −3𝑛; (38)

moreover, this bound applies for an arbitrary𝑚_𝑘.

Putting these together we have that

Pr(󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨(𝑢,∑V)∈𝐻𝑘

(𝑋𝑢V−E𝑋𝑢V)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨󵄨> 𝑎𝑘) < 2−3𝑛. (39)

This implies that

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨(𝑢,V∑)∈𝐻𝑘 𝑋_𝑢_V−E𝑋_𝑢_V 𝑝_𝑢_V|𝐴| |𝐵| 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨≤ 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨(𝑢,∑V)∈𝐻𝑘 𝑋_𝑢_V−E𝑋_𝑢_V 𝑝_𝑘−1|𝐴| |𝐵| 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 ≤ 𝑎𝑘 𝑝_𝑘−1|𝐴| |𝐵|≤ 𝛽 10( 1 √𝑒) 𝑘 , (40)

where the last two inequalities hold with probability at least

1−2−3𝑛for a given pair of sets𝐴and𝐵if𝑛is sufficiently large.

Since 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨𝑢∈𝐴,∑V∈𝐵 𝑋_𝑢_V 𝑝_𝑢_V|𝐴| |𝐵|󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨󵄨= 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 log𝑛 ∑ 𝑘=1 ∑ (𝑢,V)∈𝐻𝑘 𝑋_𝑢_V 𝑝_𝑢_V|𝐴| |𝐵| 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 ≤ log𝑛 ∑ 𝑘=1 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨(𝑢,∑V)∈𝐻𝑘 𝑋_𝑢_V 𝑝𝑘−1|𝐴| |𝐵| 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨, log𝑛 ∑ 𝑘=1 1 10( 1 √𝑒) 𝑘 ≤ 1₂, (41)

the claimed bound follows with high probability.

Remark 8. It is very similar to prove that with high probabil-ity| ∑_𝑖,𝑗𝜌_𝐺(𝑖, 𝑗) − (𝑛₂) | = 𝑜(𝑛2), we omit the details. From this it follows that rescaling the above edge weights by a factor of

(1 + 𝑜(1))and letting𝜇 ≡ 1provide us with𝛽-quasi-random

weights for most graphs from𝐺(𝑛, 𝑝_𝑖𝑗)such that𝜇(𝑉) = 𝑛and

𝜌_𝐺(𝑉, 𝑉) = 2 (𝑛

2). That is, with high probability we can apply

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4.2. Simple Examples for Defining Vertex and Edge Weights.

When defining the notion of weighted quasirandomness and

weighted regularity, we mentioned that choosing𝜇 ≡ 1and

𝜌 ≡ 1/𝛿_𝐺gives back the old definitions of quasirandomness and regularity. In the previous section we saw an example

when we needed different edge weights, but𝜇was identically

one.

Let us consider a simple example in which𝜇has to take

more than one value. Let𝐺be a star on𝑛vertices; that is,

the vertexV₁is adjacent to the verticesV₂, . . . ,V_𝑛, andV_𝑖has

degree 1 for𝑖 ≥ 2. We let𝜇(V₁) = 1/2and𝜇(V_𝑖) = 1/(2(𝑛 − 1))

for𝑖 ≥ 2and choose𝜌_𝐺≡ 𝑛/2. With these choices𝐺is easily seen to be a bipartite quasirandom; moreover, it is weighted regular.

A more sophisticated example relates weighted regularity to (𝐶, 𝜂, 𝐷) boundedness, which is the basic condition in

the regularity lemma of Alon et al. [6]. Let us recall that

𝐺 is (𝐶, 𝜂, 𝐷) bounded with parameters 𝐶 ≥ 1, 𝜂 ≥ 0

and 𝐷 is a function from𝑉 to[1, 𝑛] if for all 𝑋, 𝑌 ⊂ 𝑉,

when𝐷(𝑋), 𝐷(𝑌) ≥ 𝜂𝐷(𝑉), the inequality𝜌(𝑋, 𝑌)𝐷(𝑉) ≤

𝐶 holds, where 𝐷(𝑋) = ∑_𝑥∈𝑋𝐷(𝑥), and 𝜌(𝑋, 𝑌) :=

𝑒(𝑋, 𝑌)/(𝐷(𝑋)𝐷(𝑌)); that is,𝜌is a “generalized edge density.”

Then one can obtain an𝜀-regular partition if𝜂 ≪ 𝜀.

It is easy to check that the following graph𝐺is𝛽

-quasi-random, in fact belongs to𝐺(𝑛, 𝑝_𝑖,𝑗)with appropriate weights,

but𝐺is not(𝐶, 𝜂, 𝐷)bounded. Let𝑉(𝐺) = ∪4_𝑖=1𝐴_𝑖,|𝐴_𝑖| = 𝑛/4

for𝑖 = 1, . . . , 4. All edges between𝐴₁ and𝐴₂ are present;

there is no edge between the sets𝐴₁ ∪ 𝐴₂ and 𝐴₃ ∪ 𝐴₄,

while between𝐴₃and𝐴₄there is a random bipartite graph

with edge probability1/√𝜂. Of course, if𝜂is small enough

compared to𝐶then𝐺cannot be(𝐶, 𝜂, 𝐷)bounded.

Similarly, one can show easily that whenever a graph𝐹is

(𝐶, 𝜂, 𝐷)bounded, then with𝜇(𝑥) = 𝐷(𝑥)for all𝑥 ∈ 𝑉(𝐹)

and appropriately defined edge weights𝐹is a dense subgraph

of a graph𝐺which is(2, 𝜂)-quasi-random. Hence,Theorem 6

can be applied for𝐹. We leave the details for the reader.

4.3. Natural Weighting of𝐾_𝑛. Assume that|𝑉| = 𝑛. Let the

vertex weight function𝜇 : 𝑉 → R+ be defined such that

𝜇(𝑉) = 𝑛. We also assume that𝜇(V) = 𝑜(𝑛)for everyV ∈

𝑉, as we did earlier in the paper. Then we define thenatural

weighting of the edges of 𝐾_𝑛 (𝐾_𝑛can be replaced with other quasirandom graphs. Then the edge weights will be different.

You can find more about this at the end ofSection 4.5.)with

respect to𝜇as follows: we let𝜌(𝑢,V) = 𝜇(𝑢) ⋅ 𝜇(V)for all𝑢,V∈

𝑉,𝑢 ̸=V. Observe that

𝜌 (𝑉, 𝑉) = 𝜇 (𝑉) 𝜇 (𝑉) − ∑

V∈𝑉

𝜇2(V_{) = 2 (𝑛2)(1 − 𝑜(1)).} (42) We show that these weight functions determine a

quasiran-dom weighting of𝐾_𝑛. Let𝐴, 𝐵 ⊂ 𝑉such that𝐴 ∩ 𝐵 = 0. Then

𝜌 (𝐴, 𝐵) 𝜇 (𝐴) 𝜇 (𝐵) = ∑_𝑢∈𝐴∑_V_∈𝐵𝜇 (𝑢) 𝜇 (V) 𝜇 (𝐴) 𝜇 (𝐵) = 𝜇 (𝐴) 𝜇 (𝐵) 𝜇 (𝐴) 𝜇 (𝐵) = 1, (43)

independent of the weights of 𝐴 and 𝐵. Recalling the

definition of quasirandomness it is easy to see that the natural

weighting of𝐾_𝑛isalwaysquasirandom.

Note that natural weighting resemblesDefinition 1, where

the lower bounds on𝜇(𝐴)and𝜇(𝐵)are dropped. It is closely

related to the random model𝐺(w); see, for example, [11].

Herew = (𝑤₁, . . . , 𝑤_𝑛)is the expected degree sequence of

𝐺(w)with vertex set{1, 2, . . . , 𝑛}. The edges of𝐺(w)are drawn

independently, and the probability of including the edge𝑖𝑗

is𝑤_𝑖𝑤_𝑗/ ∑_𝑖𝑤_𝑖. Of course, the model𝐺(w)is the special case

of𝐺(𝑛, 𝑝_𝑖𝑗), andLemma 7holdswithoutany conditions. The results in the remainder of the paper also hold in more general weightings; for simplicity, we work out the details for natural weighting.

Let𝑢be an arbitrary vertex and𝐴 ⊂ 𝑉. Then theweighted

degree of 𝑢into𝐴in the graph𝐹 ⊂ 𝐾_𝑛is defined to be

𝑑𝑤_𝐹(𝑢, 𝐴) = ∑

V∈𝐴

1_𝐹(𝑢,V) 𝜇 (V) = 𝜇 (𝑁_𝐹(𝑢, 𝐴)) , ₍₄₄₎

where𝑁_𝐹(𝑢, 𝐴)denotes the neighborhood of𝑢in the set𝐴.

In particular the weighted degree of𝑢in𝐹is

𝑑𝑤_𝐹(𝑢) = ∑

V∈𝑉

1_𝐹(𝑢,V) 𝜇 (V) = 𝜇 (𝑁𝐹(𝑢)) . ₍₄₅₎

We also have that

𝜌_𝐹(𝐴, 𝐵) = ∑ 𝑢∈𝐴 ∑ V∈𝐵 1_𝐹(𝑢,V) 𝜌 (𝑢,V) = ∑ 𝑢∈𝐴 𝑑𝑤_𝐹(𝑢, 𝐵) , 𝜌𝐹(𝑉, 𝑉) = ∑ 𝑢∈𝑉𝑑𝑤𝐹(𝑢) . (46)

We define the weighted density of a weighted𝜀-regular(𝐴, 𝐵)

pair to be

𝜌_𝐹(𝐴, 𝐵)

𝜇 (𝐴) 𝜇 (𝐵). (47)

We have the following lemma.

Lemma 9. Let(𝐴, 𝐵)be a weighted𝜀-regular pair relative to the natural weighting of𝐾_𝑛with weighted density𝛾 ≫ 𝜀. Let

𝐴󸀠 _{⊂ 𝐴}_{contain only such vertices that have weighted degree}

less than(𝛾 − 𝜀)𝜇(𝐵)in the pair. Then𝜇(𝐴󸀠) < 𝜀𝜇(𝐴). Proof. Assume on the contrary that the set of “low-degree”

vertices has a large weight. Observe that𝜀-regularity implies

that

𝜌_𝐹(𝐴󸀠_{, 𝐵)}

𝜇 (𝐴󸀠_{) 𝜇 (𝐵)} > 𝛾 − 𝜀 (48)

if𝜇(𝐴󸀠) > 𝜀𝜇(𝐴). Using our assumption we get the following:

𝛾 − 𝜀 < 𝜌𝐹(𝐴 󸀠_{, 𝐵)} 𝜇 (𝐴󸀠_{) 𝜇 (𝐵)} = ∑_𝑢∈𝐴󸀠𝜇 (𝑢) 𝑑𝑤_𝐹(𝑢, 𝐵) 𝜇 (𝐴󸀠_{) 𝜇 (𝐵)} < ∑𝑢∈𝐴󸀠𝜇 (𝑢) (𝛾 − 𝜀) 𝜇 (𝐵) 𝜇 (𝐴󸀠_{) 𝜇 (𝐵)} = 𝛾 − 𝜀, (49)

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Let 𝐴, 𝐵₁, 𝐵₂, . . . , 𝐵_𝑘 be disjoint subsets of 𝑉(𝐹), and

assume that(𝐴, 𝐵_𝑖)is a weighted𝜀-regular pair relative to a

natural weighting of𝐾_𝑛 with weighted density at least𝛾for

every𝑖. Set𝛿 = 𝛾 − 𝜀. Let0 < 𝑠be an integer constant, and

assume that𝛿𝑠≫ 𝜀.

Lemma 10. Assume that𝐴󸀠⊂ 𝐴with𝜇(𝐴󸀠) > 2𝑘𝜀𝜇(𝐴). Then there exist vertices𝑢₁, 𝑢₂, . . . , 𝑢_𝑠∈ 𝐴󸀠such that

𝜇 (∩_{1≤𝑖≤𝑠}𝑁_𝐹(𝑢_𝑖, 𝐵_𝑗)) ≥ 𝛿𝑠𝜇 (𝐵_𝑗) (50)

for every1 ≤ 𝑗 ≤ 𝑘.

Proof. We find the𝑢_𝑖 vertices one by one. For 𝑢₁ we have

that the weight of vertices of𝐴with weighted degree at most

𝛿𝜇(𝐵₁)is at most𝜀𝜇(𝐴)usingLemma 9. Discard these

low-degree vertices from 𝐴󸀠; then use the regularity condition

again, this time for𝐵₂. We find that the weight of vertices

hav-ing small degree into𝐵₁or𝐵₂is at most2𝜀𝜇(𝐴). Iterating this

procedure we get that the weight of vertices that do not have

large degree into at least one𝐵_𝑖set is at most𝑘𝜀𝜇(𝐴) < 𝜇(𝐴󸀠).

Pick any of the large degree vertices from𝐴󸀠; this is our choice

for𝑢₁.

Next we repeat the process for finding𝑢₂, with the

differ-ence that we look for a vertex that has large degree into the

sets𝑁_𝐹(𝑢₁, 𝐵_𝑗)for every𝑗. Since𝜇(𝑁_𝐹(𝑢₁, 𝐵_𝑗)) ≥ 𝛿𝜇(𝐵_𝑗) ≫

𝜀𝜇(𝐵_𝑗), the same procedure will work. ApplyingLemma 9we

can find many vertices in𝐴󸀠−𝑢₁such that the weighted degree

of all of them into𝐵_𝑗is at least𝛿𝜇(𝑁_𝐹(𝑢₁, 𝐵_𝑗)) ≥ 𝛿2𝜇(𝐵_𝑗)for

every𝑗. Pick any of these; this vertex is𝑢₂.

When it comes to finding𝑢_𝑖we will work with the sets

𝐴󸀠_{− {𝑢}

1, . . . , 𝑢𝑖−1}and∩𝑡≤𝑖−1𝑁𝐹(𝑢𝑡, 𝐵𝑗)for1 ≤ 𝑗 ≤ 𝑘. Using

induction it is easy to show that

𝜇 (∩_{𝑡≤𝑖−1}𝑁_𝐹(𝑢_𝑡, 𝐵_𝑗)) ≥ 𝛿𝑖−1𝜇 (𝐵_𝑗) (51)

for every𝑗. Since𝛿𝑠 ≫ 𝜀, we can iterate this procedure until

we find all the vertices𝑢₁, . . . , 𝑢_𝑠.

Assume now that there are 𝑞clusters𝑊₁, 𝑊₂, . . . , 𝑊_𝑞 ⊂

𝑉(𝐹)such that𝜇(𝑊_𝑖) = 𝑚 + 𝑜(𝑚) for all𝑖(here𝑚 ≫ 𝜀𝑛)

and all the(𝑊_𝑖, 𝑊_𝑗)pairs are weighted𝜀-regular relative to

a natural weighting of𝐾_𝑛with density at least𝛾. That is, we

have a super-clique𝐶𝑙_𝑞on𝑞clusters.

Next we construct the graph 𝐾_𝑞𝑠, a blown-up clique, as

follows. First, we have𝑞 disjoint 𝑠-element set of vertices;

this is the vertex set of𝐾𝑠_𝑞. Then we connect any two vertices

if they belong to different vertex sets. Before we state an embedding result, we need a simple lemma; the proof is left for the reader.

Lemma 11. Let(𝐴, 𝐵)be a weighted𝜀-regular pair with density

𝑑, and for some𝛼let𝐴󸀠⊂ 𝐴with𝜇(𝐴󸀠) ≥ 𝛼𝜇(𝐴)and𝐵󸀠⊂ 𝐵 with𝜇(𝐵󸀠) ≥ 𝛼𝜇(𝐵). Then(𝐴󸀠, 𝐵󸀠)is a weighted𝜀󸀠-regular pair with𝜀󸀠=max{𝜀/𝛼, 2𝜀}and density𝑑󸀠≥ 𝑑 − 𝜀.

We have the following embedding lemma.

Lemma 12. Let𝛿 = 𝛾 − 2𝜀. If𝛿𝑞𝑠≫ 𝜀then𝐾_𝑞𝑠⊂ 𝐶𝑙_𝑞.

Proof. First, applyLemma 10with𝐴 = 𝑊₁and𝐵_𝑗 = 𝑊_𝑗+1for

1 ≤ 𝑗 ≤ 𝑞 − 1. We find the vertices𝑢1₁, 𝑢1₂, . . . , 𝑢1_𝑠 ∈ 𝑊₁such that

𝜇 (∩_{1≤𝑖≤𝑠}𝑁_𝐹(𝑢1_𝑖, 𝑊_𝑗)) ≥ 𝛿𝑠𝜇 (𝑊_𝑗) . (52)

Let𝑊_𝑗2 = ∩_𝑖≥1𝑁_𝐹(𝑢1_𝑖, 𝑊_𝑗); then𝜇(𝑊_𝑗2) ≥ 𝛿𝑠𝜇(𝑊_𝑗) ≫ 𝜀𝜇(𝑊_𝑗)

for every𝑗 ≥ 2.

Next let𝐴 = 𝑊₂2and𝐵_𝑗 = 𝑊_𝑗+22 for1 ≤ 𝑗 ≤ 𝑞 − 2. Using

Lemma 11we have that the new(𝐴, 𝐵_𝑗)pairs are all weighted

𝜀/𝛿𝑠_{-regular with density at least}_{𝛾 − 𝜀}_{. Hence, we can apply}

Lemma 10again and find𝑢2₁, 𝑢2₂, . . . , 𝑢2_𝑠 ∈ 𝑊₂2such that

𝜇 (∩_{1≤𝑖≤𝑠}𝑁_𝐹(𝑢2

𝑖, 𝑊𝑗2)) ≥ 𝛿𝑠𝜇 (𝑊𝑗2) ≥ 𝛿2𝑠𝜇 (𝑊𝑗) ≫ 𝜀𝜇 (𝑊𝑗)

(53) for3 ≤ 𝑗 ≤ 𝑞.

Continuing this process, in the 𝑘th step we will work

with the𝑊_𝑗𝑘 sets when applying Lemma 10. These sets are

defined recursively as follows:𝑊_𝑗𝑘= ∩_𝑖≥1𝑁_𝐹(𝑢𝑘−1_𝑖 , 𝑊_𝑗𝑘−1)and

𝜇(𝑊𝑘

𝑗) ≥ 𝛿(𝑘−1)𝑠𝜇(𝑊𝑗)for every𝑘 + 1 ≤ 𝑗 ≤ 𝑞. Moreover, the

(𝑊_𝑘𝑘, 𝑊_𝑗𝑘)pairs will be𝜀/𝛿(𝑘−1)𝑠-regular with density at least

𝛾 − 𝜀for every𝑘 + 1 ≤ 𝑗 ≤ 𝑞.

In the last step, when𝑘 = 𝑞 − 1, there are only two

sub-clusters left,𝑊_𝑞−1𝑞−1and𝑊_𝑞𝑞−1. The pair(𝑊_𝑞−1𝑞−1, 𝑊_𝑞𝑞−1)will be

weighted𝜀/𝛿(𝑞−2)𝑠-regular with density at least𝛾 − 𝜀. It is easy

to find a𝐾_𝑠,𝑠(a complete bipartite graph) in this regular pair

using Lemma 10. Clearly, we constructed the desired 𝐾_𝑞𝑠

graph.

4.4. Illustration: A Weighted Version of the Erd˝os-Stone

Theo-rem. Let𝑡_𝑞−1(𝑛)be the number of edges in the Tur´an graph

𝑇_{𝑛,𝑞−1}on𝑛vertices. That is,𝑇_{𝑛,𝑞−1}has the largest number of

edges such that it does not contain a𝐾_𝑞. It is well known that

lim 𝑛 → ∞ 𝑡_𝑞−1(𝑛) (𝑛 2) = 𝑞 − 2 𝑞 − 1. (54)

The Erd˝os-Stone theorem states that if one has at least

𝑡_𝑞−1(𝑛)+𝛾𝑛2_{edges (where}_{𝛾 > 0}_{is a constant) in a graph}_𝐹_on

𝑛vertices then𝐹has a𝐾_𝑞𝑠for any given natural number𝑠. In

this section we show a weighted version. We take a natural

weighting of 𝐾_𝑛 and prove that if the total edge weight in

𝐹 ⊂ 𝐾𝑛is large then𝐹has a large blown-up clique. We remark

that there are other results in the literature on the extremal

theory of weighted graphs; see, for example, [13] by Bondy

and Tuza and [14] by F¨uredi and K¨undgen, although the setup

of these papers is different from ours. Another version of the

Erd˝os-Stone theorem for sparse graphs can be found in [15].

Theorem 13. For all integers𝑞 ≥ 2and𝑠 ≥ 1and every𝛾 > 0 there exists an integer𝑛₀ such that the following holds. Take the natural weighting of𝐾_𝑛with vertex weight function𝜇and assume that𝜇(𝑉) = 𝑛 ≥ 𝑛₀. Let𝐹 ⊂ 𝐾_𝑛. If the total edge weight of𝐹is at least𝑡_𝑞−1(𝑛) + 𝛾𝑛2then𝐹contains𝐾_𝑞𝑠as a subgraph.

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Proof. We begin with applying the weighted regularity lemma

with parameters𝜀 ≪ min{(𝛾 − 𝜀)𝑞𝑠, 1/𝑠, 1/𝑞}and𝐿 ≫ 1/𝜀.

We get an𝜀-regular partition with clusters𝑊₀, 𝑊₁, . . . , 𝑊_ℓ.

Let us construct thereduced graph𝐹_𝑟as follows. The vertices

of𝐹_𝑟are identified by theℓnonexceptional clusters. We have

an edge between two vertices of𝐹_𝑟if the corresponding two

clusters give an𝜀-regular pair with density at least𝛾. Hence,

when we construct 𝐹_𝑟 we lose edges of 𝐹 as follows: (1)

edges that are incident with some vertex of𝑊₀,(2)edges that

connect two vertices that belong to the same nonexceptional

cluster,(3)edges that are in some irregular pair, and(4)edges

that are in regular pairs with small density.

The outline of the proof is as follows. We will show that the

loss in edge weight is small; hence,𝐹_𝑟will have many edges.

By Tur´an’s Theorem we will have a𝑞-clique in𝐹_𝑟. Then we

applyLemma 12and conclude the existence of a𝐾_𝑞𝑠in𝐹.

(1) The total weight of edges that are incident with some

vertex of𝑊₀can be estimated as follows:

𝜌_𝐹(𝑊₀, 𝑉) ≤ 𝜌 (𝑊₀, 𝑉) = ∑ 𝑤∈𝑊0 ∑ V∈𝑉𝜌 (𝑤, V) ≤ 𝜇 (𝑊₀) 𝜇 (𝑉) ≤ 𝜀𝑛2. (55)

(2) The nonexceptional clusters have weight(𝑛 − 𝜀𝑛)(1 +

𝑜(1))/ℓ. The total weight of edges inside nonexcep-tional clusters is at most

1 2_{1≤𝑖≤ℓ}∑ _𝑢∈𝑊∑_𝑖_V_∈𝑊∑_𝑖_−𝑢𝜌 (𝑢,V) =1₂ ∑ 1≤𝑖≤ℓ ∑ 𝑢∈𝑊𝑖 ∑ V∈𝑊𝑖−𝑢 𝜇 (𝑢) 𝜇 (V) ≤1₂ ∑ 1≤𝑖≤ℓ 𝜇(𝑊_𝑖)2= 𝑛_ℓ2(1 + 𝑜 (1)) . (56)

Sinceℓ ≥ 𝐿 ≫ 1/𝜀, we have that the total edge weight inside

nonexceptional clusters is less than𝜀𝑛2.

(3) Assume that(𝑊_𝑖, 𝑊_𝑗)is an irregular pair. Then

𝜌_𝐹(𝑊_𝑖, 𝑊_𝑗) ≤ ∑ 𝑢∈𝑊𝑖 ∑ V∈𝑊𝑗 𝜌 (𝑢,V) = ∑ 𝑢∈𝑊𝑖 ∑ V∈𝑊𝑗 𝜇 (𝑢) 𝜇 (V) = 𝜇 (𝑊_𝑖) 𝜇 (𝑊_𝑗) = 𝑛_ℓ2₂(1 + 𝑜 (1)) . (57)

Since the number of irregular pairs is at most𝜀ℓ2, we get that

the total edge weight in irregular pairs is at most

𝜀ℓ2𝑛_ℓ₂2(1 + 𝑜 (1)) < 2𝜀𝑛2. (58)

(4) If the density of an 𝜀-regular pair (𝑊_𝑖, 𝑊_𝑗)is small

then we have the following inequality:

𝜌_𝐹(𝑊_𝑖, 𝑊_𝑗) ≤ 𝜇 (𝑊_𝑖) 𝜇 (𝑊_𝑗) 𝛾 = 𝑛_ℓ2₂(1 + 𝑜 (1)) 𝛾. (59)

Since there can be at most(₂ℓ)pairs, the total edge weight in

low density pairs is less than2𝛾𝑛2/3.

Putting together, we get that the total weight of edges that we lose when applying the weighted regularity lemma is at most(4𝜀 + 2𝛾/3)𝑛2< 3𝛾𝑛2/4. Hence, the total edge weight in

the high-density regular pairs of𝐹_𝑟is at least𝑡_𝑞−1(𝑛) + 𝛾𝑛2/4.

The total weight of edges in a regular pair is(1 + 𝑜(1))𝑛2/ℓ2.

Assume that𝑒(𝐹_𝑟) ≤ ((𝑞−2)/(𝑞−1))(ℓ2/2); then the total edge

weight would be at most((𝑞−2)/(𝑞−1))(𝑛2/2)(1+𝑜(1)). Since

we have a larger edge weight in what is left after applying the

regularity lemma, using Tur´an’s theorem we get that𝐹_𝑟

con-tains a𝐾_𝑞. Every pair in this clique is a high-density𝜀-regular

pair; hence, we can applyLemma 12and find the blown-up

clique.

Remarks. One can arrive at the same conclusion perturbing

the edge weights a little. Let𝐾 > 1be a fixed constant.

Multi-ply the weight of the edge𝑒by any number𝑐_𝑒 ∈ [1/𝐾, 𝐾].

The resulting weighted graph will be quasirandom, and

it is an easy exercise to show that one still hasTheorem 13.

One can also show the weighted version of the Erd˝os-Stone-Simonovits theorem, a stability version of the above.

LetHbe a family of forbidden subgraphs having chromatic

number𝑞. Assume that the total edge weight in𝐹is close to

𝑡_𝑞−1(𝑛), but𝐹does not contain some graph𝐻 ∈H. Then𝐹_𝑟

the reduced graph cannot have a clique on𝑞vertices, but the

number of edges in it will be close to𝑡_𝑞−1(ℓ). This implies that

𝐹_𝑟is close to a Tur´an graph𝑇_ℓ,𝑞−1and that in turn implies that

the vertex set of𝐹can be partitioned into𝑞−1disjoint vertex

classes in the following way: the vertex classes all have weight

≈ 𝑛/(𝑞 − 1), the total weight of edges inside vertex classes is very small, and the weighted density of edges for every pair of disjoint classes is close to one.

4.5. Emphasized Sets. One cannot avoid having an

excep-tional cluster𝑊₀ when applying the regularity lemma. That

is, a linear number of vertices could be discarded in certain cases; a well-known example is the so-called half-graph. In general we do not have control on what is put into the excep-tional cluster. However, using vertex weights one can at least partly control the set of discarded vertices. In what follows we

show how to use the natural weighting of𝐾_𝑛in order to have

that the majority of some given emphasized set is put into nonexceptional clusters after applying the weighted regularity

lemma, even if the set is of size𝑜(𝑛). In fact we will do it for

several emphasized sets at the same time. Notice that applying

the usual regularity concept (even that of [6]) one may

discard all vertices with small degrees.

Assume that𝑘is a fixed constant and𝑉is partitioned into

the disjoint sets𝑆₁, 𝑆₂, . . . , 𝑆_𝑘, and let𝑛 = |𝑉|. Further assume

that𝑠_𝑖 → ∞as𝑛 → ∞. Let𝑠_𝑖= |𝑆_𝑖|for every𝑖. Define the

following weighting of the vertices of𝑉: forV∈ 𝑆_𝑖we let

𝜇 (V) = 𝜇𝑖= _𝑘𝑠𝑛 𝑖. (60) Observe that ∑ V∈𝑆𝑖 𝜇 (V) = 𝑛_𝑘; ₍₆₁₎

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thus, the total weight of the vertices is𝑛. LetV∈ 𝑆_𝑖and𝑤 ∈ 𝑆_𝑗.

The weight of the pair(V, 𝑤)is

𝜌 (V, 𝑤) = 𝜌𝑖𝑗= 𝜇 (V) 𝜇 (𝑤) = 𝑛 2

𝑘2_𝑠

𝑖𝑠𝑗. (62)

We showed above that 𝐾_𝑛 equipped with such vertex and

edge weights is a quasirandom graph. We call this particular

weighting thenatural weighting of 𝐾_𝑛with emphasized sets

𝑆₁, 𝑆₂, . . . , 𝑆_𝑘.

We can applyTheorem 6for some𝐹relative to the natural

weighting of𝐾_𝑛. Choose𝜀so that𝑘 ≪ 1/𝜀. Since𝜇(𝑊₀) ≤

𝜀𝑛 ≪ 𝑛/𝑘, we get that for all𝑖the majority of the vertices of

𝑆_𝑖are in nonexceptional clusters.

We remark that it is possible to define vertex weights not

only for 𝐺 = 𝐾_𝑛 but also for much sparser quasirandom

graphs when emphasizing subsets of𝑉. For example, assume

that𝐺 ∈ 𝐺(𝑛, 𝑝_𝑖𝑗), and𝑉is partitioned into the disjoint sets

𝑆₁, 𝑆₂, . . . , 𝑆_𝑘. Then one will have the vertex weights of the above example, but the edge weights will be different:

𝜌 (V_𝑖,V_𝑗) = 𝜇 (V_𝑖) 𝜇 (V_𝑗)_𝑝1

𝑖𝑗 =

𝑛2

𝑘2_𝑠

𝑞𝑠𝑡𝑝𝑖𝑗 (63)

wheneverV_𝑖 ∈ 𝑆_𝑞 andV_𝑗 ∈ 𝑆_𝑡. We leave the details for the

reader.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank P´eter Hajnal and Endre Szemer´edi for the helpful discussions. This work was partially supported by the European Union and the European Social Fund through

project FuturICT (Grant no.: T ´

AMOP-4.2.2.C-11/1/KONV-2012-0013). The first author was also supported by the ERC-AdG. 321104.

References

[1] E. Szemer´edi, “On sets of integers containing no𝑘elements in

arithmetic progression,”Acta Arithmetica, vol. 27, pp. 199–245,

1975.

[2] E. Szemer´edi, “Regular partitions of graphs,” in Probl`emes

Combinatoires et Th´eorie des Graphes, vol. 260, pp. 399–401, CNRS, Orsay, France, 1978.

[3] J. Koml´os and M. Simonovits, “Szemer´edi’s regularity lemma

and its applications in graph theory,” inCombinatorics, Paul

Erd˝os is Eighty, vol. 2, pp. 295–352, J´anos Bolyai Mathematical Society, Keszthely, Hungary, 1996.

[4] Y. Kohayakawa, “Szemer´edi’s regularity lemma for sparse

graphs,” inFoundations of Computational Mathematics, pp. 216–

230, Springer, Berlin, Germany, 1997.

[5] Y. Kohayakawa, T. Łuczak, and V. R¨odl, “Arithmetic

pro-gressions of length three in subsets of a random set,” Acta

Arithmetica, vol. 75, no. 2, pp. 133–163, 1996.

[6] N. Alon, A. Coja-Oghlan, H. H`an, M. Kang, V. R¨odl, and M. Schacht, “Quasi-randomness and algorithmic regularity for

graphs with general degree distributions,”SIAM Journal on

Computing, vol. 39, no. 6, pp. 2336–2362, 2010.

[7] A. Scott, “Szemer´edi’s regularity lemma for matrices and sparse

graphs,”Combinatorics, Probability and Computing, vol. 20, no.

3, pp. 455–466, 2011.

[8] B. Csaba and A. Pluh´ar, “Weighted regularity lemma with

applications,”http://arxiv.org/abs/0907.0245.

[9] T. Tao, “Structure and randomness in combinatorics,” in

Pro-ceedings of the 48th Annual Symposium on Foundations of Computer Science (FOCS ’07), 2007.

[10] B. Bollob´as, Random Graphs, vol. 73 of Cambridge Studies

in Advanced Mathematics, Cambridge University Press, Cam-bridge, UK, 2nd edition, 2001.

[11] F. Chung and L. Lu, “The average distance in a random graph

with given expected degrees,”Internet Mathematics, vol. 1, no. 1,

pp. 91–113, 2003.

[12] N. Alon and J. H. Spencer,The Probabilistic Method,

Wiley-Interscience, New York, NY, USA, 2nd edition, 2000.

[13] J. A. Bondy and Zs. Tuza, “A weighted generalization of Tur´an’s

theorem,”Journal of Graph Theory, vol. 25, no. 4, pp. 267–275,

1997.

[14] Z. Füredi and A. Kündgen, “Turán problems for

integer-weighted graphs,”Journal of Graph Theory, vol. 40, no. 4, pp.

195–225, 2002.

[15] D. Conlon, J. Fox, and Y. Zhao, “Extremal results in sparse

pseudorandom graphs,”Advances in Mathematics, vol. 256, pp.

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