Research Article
A Weighted Regularity Lemma with Applications
Béla Csaba
1and András Pluhár
21Bolyai Institute, University of Szeged, Aradi v´ertan´uk tere 1, Szeged 6724, Hungary
2Department of Computer Science, University of Szeged, ´Arp´ad t´er 2, Szeged 6720, Hungary
Correspondence should be addressed to B´ela Csaba; bcsaba@math.u-szeged.hu
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)
𝐴𝐵 −𝑒𝐺(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
𝑒𝐹(𝐴, 𝐵)
𝑒𝐺(𝐴, 𝐵) −𝑒𝐹(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 graph𝐾𝐴,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
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.
Set𝛿𝐺 = 𝑒(𝐺)/ (𝑛2); this is thedensity of𝐺. We define the density of the𝐴, 𝐵pair of subsets of𝑉(𝐺) by𝛿𝐺(𝐴, 𝐵) = 𝑒𝐺(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.1𝐻: (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 : (𝑉2) → R+. For𝐴, 𝐵 ⊂ 𝑉let
𝜌𝐺(𝐴, 𝐵) := ∑
𝑢∈𝐴,V∈𝐵
1𝐺(𝑢,V) 𝜌 (𝑢,V) . (4) In particular,𝜌𝐺(𝑢,V) =1𝐺(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
𝜌𝐺(𝐴, 𝐵)
𝜇 (𝐴) 𝜇 (𝐵)− 𝜌𝐺(𝑉, 𝑉)
𝜇 (𝑉) 𝜇 (𝑉) < 𝛽. (5) Observe that choosing𝜇 ≡ 1and𝜌 ≡ 1/𝛿𝐺gives 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
𝜌𝐺(𝑉, 𝑉)
𝜇 (𝑉) 𝜇 (𝑉)≤ 𝜌𝐺(𝐴, 𝐵)
𝜇 (𝐴) 𝜇 (𝐵) ≤ 𝐾 𝜌𝐺(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 := 𝜌𝐺(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)− 𝜌𝐹(𝐴, 𝐵) 𝜇 (𝐴) 𝜇 (𝐵)
< 𝜖 (7) for every𝐴 ⊂ 𝐴and𝐵 ⊂ 𝐵provided that𝜇(𝐴) ≥ 𝜖𝜇(𝐴), 𝜇(𝐵) ≥ 𝜖𝜇(𝐵). Here
𝜌𝐹(𝐴, 𝐵) = ∑
𝑢∈𝐴,V∈𝐵
1𝐹(𝑢,V) 𝜌 (𝑢,V) . (8)
Remarks. Note that weighted𝜖-regularity is nothing but the well-known𝜖-regularity when𝐺 = 𝐾𝐴,𝐵 and𝜇 ≡ 1and𝜌 is chosen to be identically the reciprocal of the density of𝐺 as before. Since1𝐹(𝑢,V) ≤ 1𝐺(𝑢,V) ≤ 1(𝑢,V)we also have 𝜌𝐹(𝐴, 𝐵) ≤ 𝜌𝐺(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, . . . , 𝑊ℓsuch that
(i)𝜇(𝑊0) ≤ 𝜖𝜇(𝑉),
(ii)|𝜇(𝑊𝑖)−𝜇(𝑊𝑗)| ≤max𝑥∈𝑉{𝜇(𝑥)}for every1 ≤ 𝑖,𝑗 ≤ ℓ, (iii) all but at most𝜖ℓ2of the pairs(𝑊𝑖, 𝑊𝑗)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
𝑐𝑖𝑠𝑖 (9)
with𝑠𝑖 ∈ 𝑆and𝑐𝑖 ∈ [−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 = 𝑀𝐽,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 : (𝑉2) → R, endowed with the inner product
⟨𝑔, ℎ⟩ = 1 (𝑛2) ∑
(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‖1𝐹‖ ≤ 1. We let𝑆be the collection of 0,1-valued functions𝛾𝐴,𝐵for𝐴, 𝐵 ⊂ 𝑉(𝐺),𝐴∩𝐵 = 0, where𝛾𝐴,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, . . . , 𝑊ℓsuch that𝐿 ≤ ℓ ≤ 𝐶𝜀,𝐿for some constant𝐶𝜀,𝐿.
Proof. Let us apply Theorem 5 to the function 1𝐹 with parameters𝜂and function𝐽to be chosen later. We get the decomposition
1𝐹= 𝑓str+ 𝑓psd+ 𝑓err, (11) where 𝑓str is (𝑀, 𝑀)-structured, 𝑓psd is 1/𝐽(𝑀)- pseudorandom, and‖𝑓err‖ ≤ 𝜂with𝑀 = 𝑀𝐽,𝜂= 𝑀𝐽,𝜀.
The function𝑓stris the combination of at most𝑀basic functions:
𝑓str= ∑
1≤𝑘≤𝑀
𝛼𝑘𝛾A𝑘,B𝑘, (12) whereA𝑘,B𝑘 are subsets of𝑉and 𝛾A𝑘,B𝑘 agrees with the indicator function of the edges of 𝐺 in between A𝑘 and B𝑘. Any(A𝑘,B𝑘)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, . . . , 𝑊ℓ, withℓ = (𝐿 + 4𝑀)/𝜀. We refer to the sets𝑊𝑖 for𝑖 = 1, . . . , ℓasclusters. If𝑛is sufficiently large then this partitioning is nontrivial. From the construction it follows that each𝑊𝑖is entirely contained within an atom. It is also clear that𝜇(𝑊0) ≤ 𝜀𝑛and𝜇(𝑊𝑖) ≈ 𝑚 = Θ(𝑛/ℓ)for every 1 ≤ 𝑖 ≤ ℓ.
We have that
𝑓err2= 1 (𝑛2) ∑
(𝑢,V)∈( 𝑉2)𝑓err(𝑢,V)2𝜌𝐺(𝑢,V) ≤ 𝜂2. (13) From this and the normalization of𝜌it follows that
1 (ℓ2) ∑
1≤𝑖<𝑗≤ℓ
1
𝜌𝐺(𝑊𝑖, 𝑊𝑗) ∑
𝑢∈𝑊𝑖,V∈𝑊𝑗
𝑓err(𝑢,V)2𝜌𝐺(𝑢,V)
= 𝑂 (𝜂2) .
(14)
Clearly, 1
𝜌𝐺(𝑊𝑖, 𝑊𝑗) ∑
𝑢∈𝑊𝑖,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∈𝑊𝑖,V∈𝑊𝑗𝑎 (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) √𝜌𝐺(𝑊𝑖, 𝑊𝑗), (17)
we get that 1
𝜌𝐺(𝑊𝑖, 𝑊𝑗) ∑
𝑢∈𝑊𝑖,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⟩ = 1 (𝑛2)
∑
𝑢∈𝐴,V∈𝐵
𝑓psd(𝑢,V) 𝜌𝐺(𝑢,V)
≤ 1 𝐽 (𝑀)
(19) for every𝐴 ⊂ 𝑊𝑖and𝐵 ⊂ 𝑊𝑗.
We will show that every good pair is weighted𝜀-regular in𝐹relative to𝐺. Let(𝑖, 𝑗)be a good pair, and assume that
𝐴 ⊂ 𝑊𝑖,𝜇(𝐴) > 𝜀𝜇(𝑊𝑖)and𝐵 ⊂ 𝑊𝑗,𝜇(𝐵) > 𝜀𝜇(𝑊𝑗). To show that(𝑊𝑖, 𝑊𝑗)is weighted𝜀-regular, it is sufficient to show that
𝜌𝐹(𝐴, 𝐵)
𝜇 (𝐴) 𝜇 (𝐵)− 𝜌𝐹(𝑊𝑖, 𝑊𝑗) 𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗)
< 𝜀. (20) Recall that
𝜌𝐹(𝐴, 𝐵) = ∑
𝑢∈𝐴,V∈𝐵
1𝐹(𝑢,V) 𝜌 (𝑢,V)
= ∑
𝑢∈𝐴,V∈𝐵
1𝐹(𝑢,V) 𝜌𝐺(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∈𝑊𝑖,V∈𝑊𝑗𝑓str(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∈𝑊𝑖,V∈𝑊𝑗𝑓psd(𝑢,V) 𝜌𝐺(𝑢,V) 𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗)
< 𝜀/3,
(23)
∑𝑢∈𝐴,V∈𝐵𝑓err(𝑢,V) 𝜌𝐺(𝑢,V)
𝜇 (𝐴) 𝜇 (𝐵) −∑𝑢∈𝑊𝑖,V∈𝑊𝑗𝑓err(𝑢,V) 𝜌𝐺(𝑢,V) 𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗)
< 𝜀/3.
(24) For proving (22) recall that𝑓stris constant on𝑊𝑖×𝑊𝑗and (𝑀, 𝑀)-structured. Since the𝛾𝑋,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∈𝑊𝑖,V∈𝑊𝑗𝑓psd(𝑢,V) 𝜌𝐺(𝑢,V) 𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗)
= (𝑛2)⟨𝑓psd, 𝛾𝑊𝑖,𝑊𝑗⟩
≤ (𝑛2)
𝐽 (𝑀) 𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗). (26)
Noting that𝜇(𝑊𝑘) = 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 (𝑊𝑖) 𝜇 (𝑊𝑗)+𝜌𝐺(𝑊𝑖, 𝑊𝑗) 𝜇 (𝐴) 𝜇 (𝐵))
≤ 𝑂 (√𝜂) 𝜌𝐺(𝑊𝑖, 𝑊𝑗) 𝜀2𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗).
(28)
We also have that
𝜌𝐺(𝑊𝑖, 𝑊𝑗)
𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗) = 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 𝐾𝑛 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,𝑝𝑖𝑗)Model. In this section we will prove that random graphs of the𝐺(𝑛, 𝑝𝑖𝑗)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, 𝑝𝑖𝑗) 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 𝑥𝑖 and 𝑥𝑗 with probability 𝑝𝑖𝑗, randomly and independently of each other. Note that if𝑝𝑖𝑗 ≡ 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, 𝑝𝑖,𝑗)is somewhat harder.
Lemma 7. Let𝛽 > 0. There exists a𝐾 = 𝐾(𝛽)such that if𝐺 ∈ 𝐺(𝑛, 𝑝𝑖𝑗)and𝑝𝑖𝑗≥ 𝐾/𝑛for every𝑖and𝑗, then𝐺is weighted𝛽- quasi-random with probability at least1−2−𝑛if𝑛is sufficiently large.
Proof. First of all let𝜇 ≡ 1, and let𝜌(𝑖, 𝑗) = 1/𝑝𝑖,𝑗. Set𝐾 = 4800/𝛽6. Let𝑝0 = 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, . . . , 𝐻𝑙: if𝑝𝑘 ≤ 𝑝𝑢V < 𝑝𝑘+1 then(𝑢,V)will belong to𝐻𝑘. Let𝑎𝑘= (𝛽3/10)√𝑒𝑘𝐾𝑛. We will denote|𝐻𝑘|by𝑚𝑘.
We will prove that the following inequality holds with probability at least1 − 2−3𝑛:
∑
𝑢∈𝐴,V∈𝐵
𝑋𝑢V 𝑝𝑢V|𝐴| |𝐵|− 1
< 𝛽/2, (30) where 𝑋𝑢V 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
∑
𝑢∈𝐴,V∈𝐵
E𝑋𝑢V
𝑝𝑢V|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 of𝐻𝑘in case𝑚𝑘is sufficiently large as follows. According to𝐴.1.11we have that
Pr( ∑
(𝑢,V)∈𝐻𝑘
(𝑋𝑢V−E𝑋𝑢V) > 𝑎𝑘) < 𝑒−(𝑎𝑘2/2𝑞𝑘𝑚𝑘)+(𝑎𝑘3/2𝑞2𝑘𝑚2𝑘), (32) where
𝑝𝑘≤ 𝑞𝑘 = ∑
(𝑢,V)∈𝐻𝑘
𝑝𝑢V
𝑚𝑘 < 𝑝𝑘+1. (33) We estimate the exponent in case𝑚𝑘= 𝑛2:
− 𝑎𝑘2
2𝑞𝑘𝑚𝑘 + 𝑎3𝑘
2𝑞2𝑘𝑚2𝑘 ≤ −𝛽6 200(√𝑒2
𝑒 )
𝑘𝐾𝑛3 𝑒𝑚𝑘 + 𝛽9
2000(√𝑒3 𝑒2 )
𝑘𝑒𝐾𝑛5 𝑚2𝑘 < −3𝑛,
(34)
where we used the definition of𝐾. For𝑚𝑘 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(𝑛
2
∑
𝑖=1𝑌𝑖>𝑎𝑘
2) , (35) where Pr(𝑌𝑖 = 1 − 𝑞𝑘) = 𝑞𝑘and Pr(𝑌𝑖 = −𝑞𝑘) = 1 − 𝑞𝑘. This implies that the exponent is at most−3𝑛even in case𝑚𝑘< 𝑛2. Indeed, let𝐴,𝐵, and𝐶be the events that∑𝑚𝑖=1𝑘𝑌𝑖 > 𝑎𝑘,
∑𝑛𝑖=12 𝑌𝑖 > 𝑎𝑘/2, and∑𝑛𝑖=𝑚2 𝑘+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( 𝑛
2
∑
𝑖=𝑚𝑘+1
𝑌𝑖< −𝑎𝑘
2) < 𝑒−𝑎2𝑘/8𝑞𝑘(𝑛2−𝑚𝑘)< 1
2. (36) With this we have proved that the sum of the weights of the edges of𝐻𝑘 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 over𝐻𝑘’s:
Pr( ∑
(𝑢,V)∈𝐻𝑘
(𝑋𝑢V−E𝑋𝑢V) < −𝑎𝑘) < 𝑒−𝑎2𝑘/2𝑞𝑘𝑚𝑘. (37) The exponent in the inequality can be estimated very similarly as before:
− 𝑎2𝑘
2𝑞𝑘𝑚𝑘 ≤ −𝛽6 200(√𝑒2
𝑒 )
𝑘𝐾𝑛3
𝑒𝑚𝑘 < −3𝑛; (38) moreover, this bound applies for an arbitrary𝑚𝑘.
Putting these together we have that Pr(
∑
(𝑢,V)∈𝐻𝑘
(𝑋𝑢V−E𝑋𝑢V)
> 𝑎𝑘) < 2−3𝑛. (39) This implies that
∑
(𝑢,V)∈𝐻𝑘
𝑋𝑢V−E𝑋𝑢V 𝑝𝑢V|𝐴| |𝐵|
≤ ∑
(𝑢,V)∈𝐻𝑘
𝑋𝑢V−E𝑋𝑢V 𝑝𝑘−1|𝐴| |𝐵|
≤ 𝑎𝑘
𝑝𝑘−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
𝑋𝑢V 𝑝𝑢V|𝐴| |𝐵|
=
log𝑛
∑
𝑘=1
∑
(𝑢,V)∈𝐻𝑘
𝑋𝑢V 𝑝𝑢V|𝐴| |𝐵|
≤
log𝑛
∑
𝑘=1
∑
(𝑢,V)∈𝐻𝑘
𝑋𝑢V 𝑝𝑘−1|𝐴| |𝐵|
,
log𝑛
∑
𝑘=1
1 10( 1
√𝑒)𝑘≤ 1 2,
(41)
the claimed bound follows with high probability.
Remark 8. It is very similar to prove that with high probabil- ity| ∑𝑖,𝑗𝜌𝐺(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, 𝑝𝑖𝑗)such that𝜇(𝑉) = 𝑛and 𝜌𝐺(𝑉, 𝑉) = 2 (𝑛2). That is, with high probability we can apply the regularity lemma for any𝐹 ⊂ 𝐺, where𝐺 ∈ 𝐺(𝑛, 𝑝𝑖𝑗).
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/𝛿𝐺gives 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, . . . ,V𝑛, andV𝑖has degree 1 for𝑖 ≥ 2. We let𝜇(V1) = 1/2and𝜇(V𝑖) = 1/(2(𝑛 − 1)) for𝑖 ≥ 2and choose𝜌𝐺≡ 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, 𝑝𝑖,𝑗)with appropriate weights, but𝐺is not(𝐶, 𝜂, 𝐷)bounded. Let𝑉(𝐺) = ∪4𝑖=1𝐴𝑖,|𝐴𝑖| = 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 of𝐾𝑛. 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 𝐾𝑛 (𝐾𝑛can 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 of𝐾𝑛. 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 of𝐾𝑛isalwaysquasirandom.
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, . . . , 𝑤𝑛)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 is𝑤𝑖𝑤𝑗/ ∑𝑖𝑤𝑖. Of course, the model𝐺(w)is the special case of𝐺(𝑛, 𝑝𝑖𝑗), 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 ⊂ 𝐾𝑛is defined to be
𝑑𝑤𝐹(𝑢, 𝐴) = ∑
V∈𝐴
1𝐹(𝑢,V) 𝜇 (V) = 𝜇 (𝑁𝐹(𝑢, 𝐴)) , (44) where𝑁𝐹(𝑢, 𝐴)denotes the neighborhood of𝑢in the set𝐴.
In particular the weighted degree of𝑢in𝐹is 𝑑𝑤𝐹(𝑢) = ∑
V∈𝑉
1𝐹(𝑢,V) 𝜇 (V) = 𝜇 (𝑁𝐹(𝑢)) . (45) We also have that
𝜌𝐹(𝐴, 𝐵) = ∑
𝑢∈𝐴
∑
V∈𝐵
1𝐹(𝑢,V) 𝜌 (𝑢,V)
= ∑
𝑢∈𝐴
𝑑𝑤𝐹(𝑢, 𝐵) , 𝜌𝐹(𝑉, 𝑉) = ∑
𝑢∈𝑉𝑑𝑤𝐹(𝑢) .
(46)
We define the weighted density of a weighted𝜀-regular(𝐴, 𝐵) pair to be
𝜌𝐹(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 of𝐾𝑛with weighted density𝛾 ≫ 𝜀. Let 𝐴 ⊂ 𝐴contain 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
𝜌𝐹(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∈𝐴𝜇 (𝑢) 𝑑𝑤𝐹(𝑢, 𝐵) 𝜇 (𝐴) 𝜇 (𝐵)
< ∑𝑢∈𝐴𝜇 (𝑢) (𝛾 − 𝜀) 𝜇 (𝐵)
𝜇 (𝐴) 𝜇 (𝐵) = 𝛾 − 𝜀,
(49)
which is clearly a contradiction.
Let 𝐴, 𝐵1, 𝐵2, . . . , 𝐵𝑘 be disjoint subsets of 𝑉(𝐹), and assume that(𝐴, 𝐵𝑖)is a weighted𝜀-regular pair relative to a natural weighting of𝐾𝑛 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, . . . , 𝑢𝑠∈ 𝐴such that
𝜇 (∩1≤𝑖≤𝑠𝑁𝐹(𝑢𝑖, 𝐵𝑗)) ≥ 𝛿𝑠𝜇 (𝐵𝑗) (50) for every1 ≤ 𝑗 ≤ 𝑘.
Proof. We find the𝑢𝑖 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 one𝐵𝑖set 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 sets𝑁𝐹(𝑢1, 𝐵𝑗)for every𝑗. Since𝜇(𝑁𝐹(𝑢1, 𝐵𝑗)) ≥ 𝛿𝜇(𝐵𝑗) ≫ 𝜀𝜇(𝐵𝑗), the same procedure will work. ApplyingLemma 9we can find many vertices in𝐴−𝑢1such that the weighted degree of all of them into𝐵𝑗is at least𝛿𝜇(𝑁𝐹(𝑢1, 𝐵𝑗)) ≥ 𝛿2𝜇(𝐵𝑗)for every𝑗. Pick any of these; this vertex is𝑢2.
When it comes to finding𝑢𝑖we will work with the sets 𝐴− {𝑢1, . . . , 𝑢𝑖−1}and∩𝑡≤𝑖−1𝑁𝐹(𝑢𝑡, 𝐵𝑗)for1 ≤ 𝑗 ≤ 𝑘. Using induction it is easy to show that
𝜇 (∩𝑡≤𝑖−1𝑁𝐹(𝑢𝑡, 𝐵𝑗)) ≥ 𝛿𝑖−1𝜇 (𝐵𝑗) (51) for every𝑗. Since𝛿𝑠 ≫ 𝜀, we can iterate this procedure until we find all the vertices𝑢1, . . . , 𝑢𝑠.
Assume now that there are 𝑞clusters𝑊1, 𝑊2, . . . , 𝑊𝑞 ⊂ 𝑉(𝐹)such that𝜇(𝑊𝑖) = 𝑚 + 𝑜(𝑚) for all𝑖(here𝑚 ≫ 𝜀𝑛) and all the(𝑊𝑖, 𝑊𝑗)pairs are weighted𝜀-regular relative to a natural weighting of𝐾𝑛with density at least𝛾. That is, we have a super-clique𝐶𝑙𝑞on𝑞clusters.
Next we construct the graph 𝐾𝑞𝑠, a blown-up clique, as follows. First, we have𝑞 disjoint 𝑠-element set of vertices;
this is the vertex set of𝐾𝑠𝑞. 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≫ 𝜀then𝐾𝑞𝑠⊂ 𝐶𝑙𝑞.
Proof. First, applyLemma 10with𝐴 = 𝑊1and𝐵𝑗 = 𝑊𝑗+1for 1 ≤ 𝑗 ≤ 𝑞 − 1. We find the vertices𝑢11, 𝑢12, . . . , 𝑢1𝑠 ∈ 𝑊1such that
𝜇 (∩1≤𝑖≤𝑠𝑁𝐹(𝑢1𝑖, 𝑊𝑗)) ≥ 𝛿𝑠𝜇 (𝑊𝑗) . (52) Let𝑊𝑗2 = ∩𝑖≥1𝑁𝐹(𝑢1𝑖, 𝑊𝑗); then𝜇(𝑊𝑗2) ≥ 𝛿𝑠𝜇(𝑊𝑗) ≫ 𝜀𝜇(𝑊𝑗) for every𝑗 ≥ 2.
Next let𝐴 = 𝑊22and𝐵𝑗 = 𝑊𝑗+22 for1 ≤ 𝑗 ≤ 𝑞 − 2. Using Lemma 11we have that the new(𝐴, 𝐵𝑗)pairs are all weighted 𝜀/𝛿𝑠-regular with density at least𝛾 − 𝜀. Hence, we can apply Lemma 10again and find𝑢21, 𝑢22, . . . , 𝑢2𝑠 ∈ 𝑊22such that
𝜇 (∩1≤𝑖≤𝑠𝑁𝐹(𝑢2𝑖, 𝑊𝑗2)) ≥ 𝛿𝑠𝜇 (𝑊𝑗2) ≥ 𝛿2𝑠𝜇 (𝑊𝑗) ≫ 𝜀𝜇 (𝑊𝑗) (53) for3 ≤ 𝑗 ≤ 𝑞.
Continuing this process, in the 𝑘th step we will work with the𝑊𝑗𝑘 sets when applying Lemma 10. These sets are defined recursively as follows:𝑊𝑗𝑘= ∩𝑖≥1𝑁𝐹(𝑢𝑘−1𝑖 , 𝑊𝑗𝑘−1)and 𝜇(𝑊𝑗𝑘) ≥ 𝛿(𝑘−1)𝑠𝜇(𝑊𝑗)for every𝑘 + 1 ≤ 𝑗 ≤ 𝑞. Moreover, the (𝑊𝑘𝑘, 𝑊𝑗𝑘)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,𝑊𝑞−1𝑞−1and𝑊𝑞𝑞−1. The pair(𝑊𝑞−1𝑞−1, 𝑊𝑞𝑞−1)will be weighted𝜀/𝛿(𝑞−2)𝑠-regular with density at least𝛾 − 𝜀. It is easy to find a𝐾𝑠,𝑠(a complete bipartite graph) in this regular pair using Lemma 10. Clearly, we constructed the desired 𝐾𝑞𝑠 graph.
4.4. Illustration: A Weighted Version of the Erd˝os-Stone Theo- rem. Let𝑡𝑞−1(𝑛)be the number of edges in the Tur´an graph 𝑇𝑛,𝑞−1on𝑛vertices. That is,𝑇𝑛,𝑞−1has the largest number of edges such that it does not contain a𝐾𝑞. It is well known that
𝑛 → ∞lim 𝑡𝑞−1(𝑛)
(𝑛2) =𝑞 − 2
𝑞 − 1. (54)
The Erd˝os-Stone theorem states that if one has at least 𝑡𝑞−1(𝑛)+𝛾𝑛2edges (where𝛾 > 0is a constant) in a graph𝐹on 𝑛vertices then𝐹has a𝐾𝑞𝑠for any given natural number𝑠. In this section we show a weighted version. We take a natural weighting of 𝐾𝑛 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 of𝐾𝑛with vertex weight function𝜇and assume that𝜇(𝑉) = 𝑛 ≥ 𝑛0. Let𝐹 ⊂ 𝐾𝑛. If the total edge weight of𝐹is at least𝑡𝑞−1(𝑛) + 𝛾𝑛2then𝐹contains𝐾𝑞𝑠as a subgraph.
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, . . . , 𝑊ℓ. Let us construct thereduced graph𝐹𝑟as follows. The vertices of𝐹𝑟are identified by theℓnonexceptional clusters. We have an edge between two vertices of𝐹𝑟if the corresponding two clusters give an𝜀-regular pair with density at least𝛾. Hence, when we construct 𝐹𝑟 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,𝐹𝑟will have many edges.
By Tur´an’s Theorem we will have a𝑞-clique in𝐹𝑟. Then we applyLemma 12and conclude the existence of a𝐾𝑞𝑠in𝐹.
(1) The total weight of edges that are incident with some vertex of𝑊0can be estimated as follows:
𝜌𝐹(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∈𝑊𝑖
∑
V∈𝑊𝑖−𝑢𝜌 (𝑢,V)
=1 2 ∑
1≤𝑖≤ℓ
∑
𝑢∈𝑊𝑖 ∑
V∈𝑊𝑖−𝑢𝜇 (𝑢) 𝜇 (V)
≤1 2 ∑
1≤𝑖≤ℓ
𝜇(𝑊𝑖)2= 𝑛2
ℓ (1 + 𝑜 (1)) .
(56)
Sinceℓ ≥ 𝐿 ≫ 1/𝜀, we have that the total edge weight inside nonexceptional clusters is less than𝜀𝑛2.
(3) Assume that(𝑊𝑖, 𝑊𝑗)is an irregular pair. Then 𝜌𝐹(𝑊𝑖, 𝑊𝑗) ≤ ∑
𝑢∈𝑊𝑖
∑
V∈𝑊𝑗
𝜌 (𝑢,V) = ∑
𝑢∈𝑊𝑖
∑
V∈𝑊𝑗
𝜇 (𝑢) 𝜇 (V)
= 𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗) = 𝑛2
ℓ2(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
𝜀ℓ2𝑛2
ℓ2 (1 + 𝑜 (1)) < 2𝜀𝑛2. (58) (4) If the density of an 𝜀-regular pair (𝑊𝑖, 𝑊𝑗)is small
then we have the following inequality:
𝜌𝐹(𝑊𝑖, 𝑊𝑗) ≤ 𝜇 (𝑊𝑖) 𝜇 (𝑊𝑗) 𝛾 = 𝑛2
ℓ2(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 of𝐹𝑟is at least𝑡𝑞−1(𝑛) + 𝛾𝑛2/4.
The total weight of edges in a regular pair is(1 + 𝑜(1))𝑛2/ℓ2. Assume that𝑒(𝐹𝑟) ≤ ((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 that𝐹𝑟con- tains a𝐾𝑞. 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 number𝑐𝑒 ∈ [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 𝑡𝑞−1(𝑛), but𝐹does not contain some graph𝐻 ∈H. Then𝐹𝑟 the reduced graph cannot have a clique on𝑞vertices, but the number of edges in it will be close to𝑡𝑞−1(ℓ). This implies that 𝐹𝑟is close to a Tur´an graph𝑇ℓ,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 of𝐾𝑛in 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, . . . , 𝑆𝑘, and let𝑛 = |𝑉|. Further assume that𝑠𝑖 → ∞as𝑛 → ∞. Let𝑠𝑖= |𝑆𝑖|for every𝑖. Define the following weighting of the vertices of𝑉: forV∈ 𝑆𝑖we let
𝜇 (V) = 𝜇𝑖= 𝑛
𝑘𝑠𝑖. (60)
Observe that
∑
V∈𝑆𝑖
𝜇 (V) = 𝑛
𝑘; (61)
thus, the total weight of the vertices is𝑛. LetV∈ 𝑆𝑖and𝑤 ∈ 𝑆𝑗. The weight of the pair(V, 𝑤)is
𝜌 (V, 𝑤) = 𝜌𝑖𝑗= 𝜇 (V) 𝜇 (𝑤) = 𝑛2
𝑘2𝑠𝑖𝑠𝑗. (62) We showed above that 𝐾𝑛 equipped with such vertex and edge weights is a quasirandom graph. We call this particular weighting thenatural weighting of 𝐾𝑛with emphasized sets 𝑆1, 𝑆2, . . . , 𝑆𝑘.
We can applyTheorem 6for some𝐹relative to the natural weighting of𝐾𝑛. Choose𝜀so that𝑘 ≪ 1/𝜀. Since𝜇(𝑊0) ≤ 𝜀𝑛 ≪ 𝑛/𝑘, we get that for all𝑖the majority of the vertices of 𝑆𝑖are in nonexceptional clusters.
We remark that it is possible to define vertex weights not only for 𝐺 = 𝐾𝑛 but also for much sparser quasirandom graphs when emphasizing subsets of𝑉. For example, assume that𝐺 ∈ 𝐺(𝑛, 𝑝𝑖𝑗), and𝑉is partitioned into the disjoint sets 𝑆1, 𝑆2, . . . , 𝑆𝑘. Then one will have the vertex weights of the above example, but the edge weights will be different:
𝜌 (V𝑖,V𝑗) = 𝜇 (V𝑖) 𝜇 (V𝑗) 1
𝑝𝑖𝑗 = 𝑛2
𝑘2𝑠𝑞𝑠𝑡𝑝𝑖𝑗 (63) wheneverV𝑖 ∈ 𝑆𝑞 andV𝑗 ∈ 𝑆𝑡. 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.
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