Cliques and Chromatic Number in Inhomogenous Random Graphs

TL;DR

This paper investigates the properties of cliques and chromatic numbers in inhomogeneous random graphs, providing bounds and estimates even when edge probabilities are very low, extending understanding of graph coloring in sparse and diverse networks.

Contribution

It introduces a recursive method to estimate maximum clique size in inhomogeneous graphs and derives uniform bounds for both inhomogeneous and homogeneous cases across various edge probabilities.

Findings

Maximum clique size estimates for inhomogeneous graphs.

Uniform bounds on clique size and chromatic number for homogeneous graphs.

Applicability across a wide range of edge probabilities.

Abstract

In this paper, we study cliques and chromatic number of inhomogenous random graphs where the individual edge probabilities could be arbitrarily low. We use a recursive method to obtain estimates on the maximum clique size under a mild positive average edge density assumption. As a Corollary, we also obtain uniform bounds on the maximum clique size and chromatic number for homogenous random graphs for all ranges of the edge probability $p_n$ satisfying $\frac{1}{n^{\alpha_1}} \leq p_n \leq 1-\frac{1}{n^{\alpha_2}}$ for some positive constants $\alpha_1$ and $\alpha_2$.

Equations

$${\mathbb{P}}_{i,j}(X(i,j)=1)=p(i,j)=1-{\mathbb{P}}_{i,j}(X(i,j)=0).$$

$$\log \left(\frac{1}{t_n}\right):=\underset{S:\#S=U_n}{\inf }{\left(\left(\genfrac{}{}{0ex}{}{\#S}{2}\right)\right)}^{-1}\sum _{i,j\in S}\log \left(\frac{1}{p(i,j)}\right).$$

$${\mathbb{P}}_{\text{p}}\left(\omega (G(n,\text{p}))\le U_n\right)\ge 1-\exp \left(-f_n{U}_n\right)$$

$$f_n:=\frac{(U_n-1)}{2}\log \left(\frac{1}{t_n}\right)-\log n$$

$$\underset{1\le i\le n}{\inf }\underset{S}{\inf }\frac{1}{\#S}\sum _{j\in S}p(i,j)\ge p_n.$$

$${\alpha }_1=\underset{n}{\mathrm{limsup}}\frac{\log \left(\frac{1}{p_n}\right)}{\log n}$$

$${\alpha }_2=\underset{n}{\mathrm{limsup}}\frac{\log \left(\frac{1}{1-p_n}\right)}{\log n}.$$

$$\max \left(\frac{{\alpha }_1}{2},a\right)+\gamma <\eta <1.$$

$${\mathbb{P}}_{\text{p}}\left(\omega (G(n,\text{p}))\ge (1-\eta )\frac{\log n}{\log \left(\frac{1}{p_n}\right)}\right)\ge 1-3\exp \left(-n^{2\eta -2\gamma -{\alpha }_1}\right)$$

$$a<\gamma <\eta <1.$$

$${\mathbb{P}}_{\text{p}}\left(\omega (G(n,\text{p})\ge (1-\eta )\frac{\log n}{\log \left(\frac{1}{p_n}\right)}\right)\ge 1-3\exp \left(-n^{2\eta -2\gamma }\right)$$

$$\max \left(\gamma -\frac{{\alpha }_2}{2},a\right)<\eta <1-{\alpha }_2.$$

$${\mathbb{P}}_{\text{p}}\left(\omega (G(n,\text{p}))\ge (1-{\alpha }_2-\eta )\frac{\log n}{\log \left(\frac{1}{p_n}\right)}\right)\ge 1-3\exp \left(-n^{2\eta -2\gamma +{\alpha }_2}\right)$$

$$\mathbb{P}\left(\omega (G(n,p_n))=1\right)\ge 1-\frac{1}{n^{{\alpha }_1-ϵ-2}}$$

$$\mathbb{P}\left(\omega (G(n,p_n))=n\right)\ge 1-\frac{1}{n^{{\alpha }_2-ϵ-2}}$$

$$\mathbb{P}\left(\omega (G(n,p_n)\ge n-n^{2-{\alpha }_2+2ϵ}\right)\ge 1-\exp \left(-n^{2-{\alpha }_2-2ϵ}\right)$$

$$U_n=\frac{2\log n+2f_n}{\log \left(\frac{1}{p_n}\right)}+1\ge 1.$$

$$\mathbb{P}\left(\omega (G(n,p_n))\le U_n\right)\ge 1-\exp \left(-f_n{U}_n\right)$$

$${\mathbb{P}}_{\text{p}}\left(\frac{1-\eta }{{\theta }_1}\le \omega (G(n,p_n))\le \frac{(2+\xi )}{{\theta }_1}+1\right)\ge 1-3\exp \left(-n^{2\eta -2\gamma -{\theta }_1}\right)-n^{-\frac{\xi (2+\xi )}{{\theta }_1}}$$

$$\mathbb{P}\left(\frac{(1-\eta )\log n}{\log \left(\frac{1}{p}\right)}\le \omega (G(n,p_n))\le \frac{(2+\xi )\log n}{\log \left(\frac{1}{p}\right)}\right)$$

$$\ge 1-3\exp \left(-n^{2\eta -2\gamma }\right)-\exp \left(\frac{-\xi (1+\xi )}{\log \left(\frac{1}{p}\right)}(\log n{)}^2\right)$$

$$\mathbb{P}\left((1-{\theta }_2-\eta )n^{{\theta }_2}\log n\le \omega (G(n,p_n))\le (2+\xi )n^{{\theta }_2}\log n\right)$$

$$\ge 1-3\exp \left(-n^{2\eta -2\gamma +{\theta }_2}\right)-\exp \left(-\xi (1+\xi )n^{{\theta }_2}(\log n{)}^2\right)$$

$$\mathbb{P}\left((1-\xi )\frac{n^{1-{\theta }_2}}{2\log n}\le \chi (G(n,r_n))\le \frac{2(1+\xi )}{1-2{\theta }_2}\frac{n^{1-{\theta }_2}}{\log n}\right)$$

$$\ge 1-3\exp \left(-n^{1-{\theta }_2-\zeta }\right)-\exp \left(-\xi (1+\xi )n^{{\theta }_2}(\log n{)}^2\right)$$

$$\mathbb{P}\left((1-\xi )\frac{n\log \left(\frac{1}{1-p}\right)}{2\log n}\le \chi (G(n,r_n))\le 2(1+\xi )\frac{n\log \left(\frac{1}{1-p}\right)}{\log n}\right)$$

$$\ge 1-3\exp \left(-n^{1-\zeta }\right)-\exp \left(-\frac{\xi (1+\xi )}{\log \left(\frac{1}{1-p}\right)}(\log n{)}^2\right)$$

$$\mathbb{P}\left((1-\xi )\frac{{\theta }_{1}n}{2+{\theta }_1}\le \chi (G(n,r_n))\le (1+\xi )\frac{2{\theta }_{1}n}{1-{\theta }_1}\right)$$

$$\ge 1-3\exp \left(-n^{2\eta -2\gamma -{\theta }_1}\right)-n^{-\frac{\xi (1+\xi )}{{\theta }_1}}$$

$${\mathbb{P}}_{\text{p}}(B_L(S_n))\le \sum _{S:\#S=L}\prod _{i,j\in S}p(i,j)=\sum _{S:\#S=L}\exp \left(-\sum _{i,j\in S}\log \left(\frac{1}{p(i,j)}\right)\right).$$

$${\mathbb{P}}_{\text{p}}(B_L(S_q))$$

$$\mathbb{P}(X_i=1)=p_i=1-\mathbb{P}(X_i=0).$$

$$T_m=\sum _{i=1}^m{X}_i,$$

Full text

Key words: Random graphs, inhomogenous edge probabilities, cliques, chromatic number.

1 Introduction

Let $K_{n}$ be the labelled complete graph on $n$ vertices with vertex set

$\{1,2,\ldots,n\}$ and edge set $\{e_{1},e_{2},\ldots,e_{m}\},$ where $m={n\choose 2}.$ Let $G_{n}=G(n,p_{n})$ be the random graph obtain when every edge is independently open with probability $p_{n}\in(0,1)$ and closed otherwise. Let $X(i,j)$ be a Bernoulli random variable defined on the probability space

$(\{0,1\},\mathbb{B}(\{0,1\}),\mathbb{P}_{i,j})$ with

[TABLE]

Here $\mathbb{B}(\{0,1\})$ is the set of all subsets of $\{0,1\}.$ We say that edge $e(i,j)$ is open if $X(i,j)=1$ and closed otherwise. The random variables $\{X(i,j)\}$ are independent and the resulting random graph $G$ is an inhomogenous Erdős-Rényi (ER) random graph, defined on the probability space $(\Omega,{\cal F},\mathbb{P}).$ Here $\Omega=\{0,1\}^{{n\choose 2}},$ the sigma algebra ${\cal F}$ is the set of subsets of $\Omega$ and $\mathbb{P}=\prod_{i,j}\mathbb{P}_{i,j}.$

Clique Number

Let $G=(V,E)$ be a graph with vertex set $V=V(G)$ and edge set $E=E(G).$ Suppose that $\#V=n$ so that $G$ is a graph on $n$ vertices. We say that $G$ is a complete graph if $\#E={n\choose 2}.$ For a subset $U\subset V$ containing $\#U=r\leq n$ vertices, we define $G|_{U}=(U,E_{U})$ to be the induced subgraph of $G$ with vertex set $U,$ defined as follows. For any two vertices $a,b\in U,$ the edge $e_{ab}$ with endvertices $a$ and $b$ belongs to $E_{U}$ if and only if $e_{ab}\in E(G).$ We say that $G|_{U}$ is a clique if $G|_{U}$ is a complete graph. We denote $\omega(G)$ to be the size of the largest clique in $G.$ Throughout, the size of a graph will always refer to the number of vertices in the graph.

Let $\textbf{p}=\{p(i,j)\}_{i,j}$ be a vector of probabilities as defined in the previous subsection and let $G(n,\textbf{p})$ be the resulting random graph. The following obtains an upper bound for the clique number $\omega(G(n,\textbf{p})).$

Proposition 1**.**

For any sequence $U_{n}>0$ define

[TABLE]

We have

[TABLE]

where

[TABLE]

for all $n\geq 2.$

To obtain a lower bound on the clique number, we have a few definitions first. As before, let p be the vector formed by the probabilities $\{p(i,j)\}_{i,j}.$ For $p_{n}>0$ and constant $0\leq a<1,$ let ${\cal N}(a,n,p_{n})$ be the set of all vectors p satisfying the following condition: There is a constant $N=N(a)\geq 1$ such that for all $n\geq N,$ we have

[TABLE]

Here $N=N(a)\geq 1$ is a constant not depending on $n.$ For a fixed $1\leq i\leq n,$ the infimum above is taken over all sets $S$ such that $\#S\geq n^{a}$ and $i\notin S.$ The condition implies that the average edge density taken over sets of cardinality at least $n^{a},$ is at least $p_{n}.$ All constants mentioned throughout are independent of $n.$

Let $p_{n}$ be as in (1.4) and define

[TABLE]

and

[TABLE]

We consider three cases separately depending on whether $\alpha_{1}>0$ or $\alpha_{2}>0$ or both $\alpha_{1}=\alpha_{2}=0.$

Theorem 1**.**

*Suppose $\textbf{p}\in{\cal N}(a,n,p_{n})$ for some constant $0\leq a<1.$

$(i)$ Suppose $0<\alpha_{1}<2$ and let $\eta,\gamma>0$ be such that*

[TABLE]

We have that $\alpha_{2}=0$ and there is a positive integer $N_{1}=N_{1}(\eta,\gamma,\alpha_{1},a)\geq 1$ so that

[TABLE]

*for all $n\geq N_{1}.$

$(ii)$ Suppose $\alpha_{1}=\alpha_{2}=0.$ Let $\eta,\gamma>0$ be such that*

[TABLE]

There is a positive integer $N_{2}=N_{2}(\eta,\gamma)\geq 1$ so that

[TABLE]

*for all $n\geq N_{2}.$

$(iii)$ Suppose $0<\alpha_{2}<1$ and let $\eta,\gamma>0$ be such that*

[TABLE]

We have that $\alpha_{1}=0$ and there is a positive integer $N_{3}=N_{3}(\eta,\gamma)\geq 1$ so that

[TABLE]

for all $n\geq N_{3}.$

The usual method for studying the lower bound for clique numbers of homogenous random graphs uses a combination of second moment method and martingale inequalities (see for e.g., Alon and Spencer (2003), Bollobas (2001)). For inhomogenous graphs where the edge probabilities could be arbitrarily low, the above method is not directly applicable. We use a recursive method to obtain lower bounds on the clique numbers (see Lemma 5, Section 2).

As a consequence of our main Theorem above, we also obtain results for homogenous random graphs where the edge probabilities $p(i,j)=p_{n}$ for $1\leq i\neq j\leq n.$

Proposition 2**.**

If $\alpha_{1}>2,$ then fix $\epsilon>0$ small so that $\alpha_{1}-\epsilon>2.$ We then have

[TABLE]

for all $n$ large. Similarly, if $\alpha_{2}>2,$ then fix $\epsilon>0$ small so that $\alpha_{2}-\epsilon>2.$ We then have

[TABLE]

for all $n$ large. If $1<\alpha_{2}<2,$ then fix $\epsilon>0$ small so that $0<2-\alpha_{2}-2\epsilon<2-\alpha_{2}+2\epsilon<1.$ We then have

[TABLE]

for all $n$ large.

Let $f_{n}\rightarrow\infty$ be any sequence and let

[TABLE]

We have

[TABLE]

for all $n\geq 2.$

We have the following result regarding the clique number for the cases where $\alpha_{1}<1$ and $\alpha_{2}<1.$

Corollary 2**.**

$(i)$* Suppose $p_{n}=\frac{1}{n^{\theta_{1}}}$ for some $0<\theta_{1}<1.$ Fix $\eta,\gamma>0$ such that $\frac{\theta_{1}}{2}+\gamma<\eta<1$ and $\xi>0.$ There is a positive integer $N_{1}=N_{1}(\eta,\gamma,\xi)\geq 1$ so that*

[TABLE]

*for all $n\geq N_{1}.$

$(ii)$ Suppose $p_{n}=p\in(0,1)$ for all $n.$ Fix $0<\gamma<\eta<1$ and $\xi>0.$ There is a positive integer $N_{2}=N_{2}(\eta,\gamma,\xi)\geq 1$ so that*

[TABLE]

*for all $N\geq N_{2}.$

$(iii)$ Suppose $p_{n}=1-\frac{1}{n^{\theta_{2}}}$ for some $0<\theta_{2}<1.$ Fix $\eta,\gamma>0$ so that $\gamma-\frac{\theta_{2}}{2}<\eta<1-\theta_{2}$ and fix $\xi>0.$ There is a positive integer $N_{3}=N_{3}(\eta,\gamma,\xi)\geq 1$ so that*

[TABLE]

for all $n\geq N_{3}.$

Chromatic Number

We have the following result regarding the chromatic number for homogenous random graphs where each edge is independently open with probability $r_{n}.$ We discuss separate cases depending on the asymptotic behaviour of $r_{n}.$

Theorem 3**.**

$(i)$* Suppose $r_{n}=\frac{1}{n^{\theta_{2}}}$ for some $0<\theta_{2}<\frac{1}{2}.$ Fix $\xi,\zeta>0.$ There is a constant $N_{1}=N_{1}(\xi,\theta_{2})\geq 1$ so that*

[TABLE]

*for all $n\geq N_{1}.$

$(ii)$ Suppose $r_{n}=p$ for some $0<p<1$ and for all $n.$ Fix $\xi,\zeta>0.$ There is a constant $N_{2}=N_{2}(\xi,\zeta)\geq 1$ so that*

[TABLE]

*for all $n\geq N_{2}.$

$(iii)$ Suppose $r_{n}=1-\frac{1}{n^{\theta_{1}}}$ for some $0<\theta_{1}<1.$ Fix $\xi,\zeta>0.$ There is a constant $N_{3}=N_{3}(\xi,\zeta)\geq 1$ so that*

[TABLE]

for all $n\geq N_{3}.$

The paper is organized as follows. In Section 2, we prove Proposition 1 and obtain preliminary estimates for proving the main Theorem 1. In Section 3, we prove Theorem 1 regarding the lower bound for clique numbers of inhomogenous graphs. In Section 4, we prove Proposition 2 and Theorem 2 for clique numbers of homogenous graphs. Finally in Section 5, we prove Theorem 3 regarding the chromatic number for homogenous graphs.

2 Preliminary estimates

For integer $q\geq 1,$ let $G(q,\textbf{p})$ be the random graph with vertex set $S_{q}=\{1,2,\ldots,q\}.$ For integer $L\geq 2,$ let $B_{L}(S_{q})$ denote the event that the random graph $G(q,\textbf{p})$ contains an open $L-$clique; i.e., there are vertices $\{v_{i}\}_{1\leq i\leq L}$ such that the edge between $v_{i}$ and $v_{j}$ is open for any $1\leq i\neq j\leq L.$

Proof of Proposition 1: We have

[TABLE]

Setting $L=U_{n}$ and using the definition of $t_{n}$ in (1.16), we have

[TABLE]

where $f_{n}$ is as defined in (1.3). This proves the upper bound (1.17) in Proposition 1.

In what follows, we estimate the probability $\mathbb{P}(B^{c}_{L}(S_{q}))$ to obtain the lower bounds in Theorem 1. We use the following Binomial estimate. Let $\{X_{i}\}_{1\leq i\leq m}$ be independent Bernoulli random variables with

[TABLE]

We have the following Lemma.

Lemma 4**.**

Fix $0<\epsilon<\frac{1}{6}.$ If

[TABLE]

then

[TABLE]

for all $m\geq 1.$

For proof we refer to the Wikipedia link:

$https://en.wikipedia.org/wiki/Chernoff\_bound$.

Small cliques estimate

For integer $q\geq 1,$ we recall that $G(q,p)$ is the random graph with vertex set $S_{q}=\{1,2,\ldots,q\}.$ For integer $L\geq 2,$ let $B_{L}(S_{q})$ denote the event that the random graph $G(q,\textbf{p})$ contains an open $L-$clique; i.e., there are vertices $\{v_{i}\}_{1\leq i\leq L}$ such that the edge between $v_{i}$ and $v_{j}$ is open for any $1\leq i\neq j\leq L.$ For $L\geq 2,$ we define

[TABLE]

We first obtain a recursive relation involving $t_{L}(q).$

Lemma 5**.**

Fix $0<\epsilon<\frac{1}{6}$ and integer $L\geq 1.$ For integer $q\geq 1,$ suppose that $\textbf{p}\in{\cal N}(a,q,p_{q})$ (see (1.4)) and let

[TABLE]

be the largest integer less than or equal to $(p_{q}-\delta)(q-1).$ Let $\delta\in\{p_{q}\epsilon,(1-p_{q})\epsilon\}.$ For all integers $q$ such that $q_{1}\geq n^{a},$ we have that

[TABLE]

Proof of Lemma 5: For simplicity, we write $p=p_{q}.$ We first prove that (2.5) is satisfied with $\delta=p\epsilon.$ Let $N_{e}$ be the number of open edges in the random graph $G(q,\textbf{p}).$ Using (1.4), we have that $\mathbb{E}N_{e}\geq p{q\choose 2}.$ Fixing $0<\epsilon<\frac{1}{6}$ and applying the binomial estimate (2.2) with $T_{m}=N_{e},$ we have that

[TABLE]

for all $q\geq 2.$ The final term is obtained using $\delta=p\epsilon.$ Using $\frac{1}{4}{q\choose 2}\geq\frac{q^{2}}{10}$ for all $q\geq 5$ for the final term above we have

[TABLE]

Using (2.7), we therefore have

[TABLE]

where

[TABLE]

and

[TABLE]

We estimate $I_{1}$ as follows. Suppose that the event $N_{e}\geq(p-\delta){q\choose 2}$ occurs. If $d(v)$ denotes the degree of vertex $v\in\{1,2,\ldots,q\}$ in the random graph $G(q,\textbf{p}),$ we then have

[TABLE]

In particular, there exists a vertex $w$ such that

[TABLE]

Here $q_{1}=[(p-\delta)(q-1)]$ is as defined in the statement of the Lemma. This implies that

[TABLE]

Fixing $1\leq z\leq q,$ we evaluate each term in (2.12) separately. Letting $N(z)=N(z,G(q,\textbf{p}))$ be the set of neighbours of $z$ in the random graph $G(q,p),$ we have

[TABLE]

Suppose now that the event $B^{c}_{L}(S_{q})\bigcap\{N(z)=S\}$ occurs for some fixed set $S$ with $\#S\geq q_{1}.$ We recall that since $B_{L}^{c}(S_{q})$ occurs, there is no $L-$clique in the random graph $G(q,\textbf{p})$ with vertex set $S_{q}=\{1,2,\ldots,q\}.$ This means that there is no $(L-1)-$clique in the random induced subgraph of $G(q,\textbf{p})$ formed by the vertices of $S;$ i.e., the event $B^{c}_{L-1}(S)$ occurs. Therefore we have

[TABLE]

The equality (2.14) true as follows. The event that $\{N(z)=S\}$ depends only on the state of edges containing $z$ as an endvertex. On the other hand, the event $B^{c}_{L-1}(S)$ depends only on the state of edges having both their endvertices in $S.$ Since the set $S$ does not contain the vertex $z$ (see (2.13)), we have that the events $\{N(z)=S\}$ and $B^{c}_{L-1}(S)$ are independent. This proves (2.14).

We obtain the desired recursion using (2.14) as follows. We recall that the set $S$ contains at least $q_{1}$ vertices (see (2.13)). Therefore, setting $T$ to be the set of the $q_{1}$ least indices in $S,$ we have that if $B^{c}_{L-1}(S)$ occurs, then $B^{c}_{L-1}(T)$ occurs; i.e., there is no $(L-1)-$clique in the random induced subgraph formed by the vertices of $T.$ From (2.14), we therefore have that

[TABLE]

The final inequality is true as follows. Let $\textbf{p}_{T}$ be the vector formed by the probabilities $\{p(i,j)\}_{i,j\in T}.$ From (1.4), we then have

[TABLE]

for all $n\geq N.$ As in (1.4), the infimum is taken over all sets $S\subset T$ such that $\#S\geq n^{a}$ and $i\notin S.$ This proves that $\textbf{p}_{T}\in{\cal N}(a,q_{1},p)$ and so (2.15) is true.

Substituting (2.15) into (2.13), we have

[TABLE]

The equality (2.17) is true since the events $\{N(z)=S\}$ are disjoint for distinct $S.$ Substituting (2.18) into (2.12), we have

[TABLE]

Using estimates (2.19) and (2.10) in (2.8) gives

[TABLE]

for all $q$ such that $q_{1}\geq n^{a}.$ Taking supremum over all $\textbf{p}\in{\cal N}(a,q,p)$ proves (2.5) with $\delta=p\epsilon.$

It remains to see that (2.5) is satisfied with $\delta=\epsilon(1-p).$ We recall that $N_{e}$ denotes the number of open edges in the random graph $G(n,p)$ (see the first paragraph of this proof). Let $W_{e}={n\choose 2}-N_{e}$ denote the number of closed edges. Fixing $0<\epsilon<\frac{1}{6}$ and applying the binomial estimate (2.2) with $T_{m}=W_{e},$ we have that

[TABLE]

for all $q\geq 2.$ The final estimate follows using $\delta=\epsilon(1-p).$ Since

[TABLE]

we again obtain (2.6). The rest of the proof is as above.

We use the recursion in the above Lemma iteratively to estimate the probability $t_{L}(q)$ of the event that there is no open $L-$clique in the random graph $G(q,\textbf{p}).$

Lemma 6**.**

For integer $i\geq 1,$ define

[TABLE]

For all $q\geq 1$ such that $v_{L}(q)\geq n^{a},$ we have

[TABLE]

where

[TABLE]

and

[TABLE]

To prove the above Lemma, we have a couple of preliminary estimates. Let $\{q_{i}\}_{0\leq i\leq L}$ be integers defined recursively as follows. The term $q_{0}=q$ and for $i\geq 1,$ let

[TABLE]

For a fixed $1\leq i\leq L,$ we have the following estimates.

$(a1)$ We have

[TABLE]

$(a2)$ For $\delta>0,$ we have

[TABLE]

Proof of $(a1)-(a2)$: The property $(a1)$ is obtained using the property $x-1\leq[x]\leq x$ for any $x>0.$ Applying the upper bound in (2.25) recursively, we get

[TABLE]

This proves the upper bound in (2.26). For the lower bound we again proceed iteratively and obtain for $i\geq 2$ that

[TABLE]

Since $\delta>0,$ we have

[TABLE]

and so

[TABLE]

This proves $(a2).$

Using the properties $(a1)-(a2),$ we prove Lemma 6.

Proof of Lemma 6: Letting

[TABLE]

we apply the recursion (2.5) successively to get

[TABLE]

for all $q$ such that $q_{2}=q_{2}(q)\geq n^{a}.$ The final estimate follows since $q_{1}\leq q$ (see (2.25) of property $(a1)$). Proceeding iteratively, we obtain the following estimate for all $q$ such that $q_{L-2}(q)\geq n^{a}:$

[TABLE]

where

[TABLE]

and

[TABLE]

Let $v_{L}=v_{L}(q)$ be as defined in (2.21). For all $q$ such that $v_{L}(q)\geq n^{a},$ we have the following bounds for the terms $J_{1}$ and $J_{2}.$

[TABLE]

and

[TABLE]

where $A_{1}$ and $A_{2}$ are as given in (2.23) and (2.24), respectively. This proves the Lemma.

Proof of (2.32) and (2.33): Since

[TABLE]

for all $1\leq j\leq L-1$ (property $(a2)$), the estimate (2.29) holds for all $q$ such that $v_{L}=v_{L}(q)\geq n^{a}.$

We first evaluate $J_{1}.$ We have

[TABLE]

and

[TABLE]

The first equality in (2.36) is true since there is no open $2-$clique among a set of vertices if and only if all the edges between the vertices are closed. The second and third inequality follow from (2.34) and the fact that $1-p<1.$ Substituting (2.36) into (2.35) we get the estimate (2.32) for the term $J_{1}.$

For the second term $J_{2},$ we argue as follows. The term $r(q)=\exp\left(-\frac{\epsilon\delta}{10}q^{2}\right)$ defined in (2.28) is decreasing in $q.$ For $1\leq j\leq L-1,$ we have from (2.34) that $q_{j}\geq v_{L}$ and so $r(q_{j})\leq r(v_{L}).$ Using this in (2.31), we then have

[TABLE]

The first inequality follows from the fact that $\frac{q^{L-2}-1}{q-1}\leq 2q^{L-3}$ for all $q\geq 3.$ Using the expression for $r(q)$ in (2.37), we obtain (2.33).

3 Proof of Theorem 1

The following two estimates are used in what follows. For $0<x<1,$ we have

[TABLE]

and

[TABLE]

Proof of $(i)$

Here $\alpha_{1}>0$ and we use the estimates (2.23) and (2.24) of Lemma 6 to prove the Theorem 1. We first obtain a couple of additional estimates. Fix $\eta$ and $\gamma$ as in the statement of the Theorem. Also fix $\epsilon>0$ small to be determined later and set $q_{n}=n,$

[TABLE]

and

[TABLE]

For a fixed $\epsilon>0,$ we have the following estimates regarding $L_{n}$ and $\delta_{n}.$

$(b1)$ We have that

[TABLE]

and so

[TABLE]

$(b2)$ We have

[TABLE]

for all $n$ large.

$(b3)$ There is a constant $N_{0}=N_{0}(\eta,\epsilon)\geq 1$ such that

[TABLE]

for all $n$ large.

Proof of $(b1)-(b3)$: We prove $(b1)$ first. We use the definition of $\alpha_{1}>0$ to get that

[TABLE]

for all $n$ large. This proves (3.5) and (3.6) in property $(b1).$

The inequality in (3.7) follows from the final estimate of (3.9) and the definition of $L_{n}$ in (3.3). This proves $(b2).$ To prove property $(b3),$ we argue as follows. Setting $q=q_{n}=n$ and $L=L_{n}$ in the definition of $v_{i}$ in (2.21), we then have

[TABLE]

where

[TABLE]

The final estimate in (3.10) follows from the final estimate in (3.5). The equality (3) above is obtained using $\delta_{n}=\epsilon p_{n}$ and the final equality (3.12) follows from the definition of $L_{n}$ in (3.3).

Using $-\log(1-x)<\frac{x}{1-x}$ (see (3.1)) with $x=\epsilon,$ we have

[TABLE]

where the final estimate follows using (3.7) in property $(b2)$ above. Substituting the above into (3.12), we have

[TABLE]

for all $n$ large. Using (3.13) in (3.10), we have

[TABLE]

for all $n$ large.

We use properties $(b1)-(b3)$ to prove $(i)$ in Theorem 1.

Proof of $(i)$: From property $(b3)$ and the choices of $\eta$ and $\gamma>0$ as in the statement of the Thoerem, we have that $v_{L_{n}}\geq n^{a}$ for all $n\geq N_{1}$ large. Here $N_{1}=N_{1}(\eta,\gamma,a)$ does not depend on the choice of $\textbf{p}.$ Thus the estimates for $A_{1}$ and $A_{2}$ in Lemma 6 are applicable.

Setting $q=q_{n}=n$ and $L=L_{n}$ and $\delta=\delta_{n}$ as in (3.3) and (3.4), respectively, in the expressions for $A_{1}$ and $A_{2}$ in (2.23) and (2.24), we have

[TABLE]

and

[TABLE]

The following estimates for $A_{1}$ and $A_{2}$ imply the lower bound (1.8) for case $(i)$ in Theorem 1. Fix $\gamma,\eta>0$ as in the statement of the Theorem.

$(c1)$ There are positive constants $\epsilon=\epsilon(\eta,\gamma)>0$ and $M_{1}=M_{1}(\eta,\gamma,\alpha_{1})\geq 1$ so that

[TABLE]

for all $n\geq M_{1}.$

$(c2)$ There are positive constants $\epsilon=\epsilon(\eta,\gamma)>0$ and $M_{2}=M_{2}(\eta,\gamma,\alpha_{1})\geq 1$ so that

[TABLE]

for all $n\geq M_{2}.$

Proof of $(c1)-(c2)$: We first prove $(c1).$ Using the estimate (3.7) of property $(b2),$ we have that the first term in (3.15) is

[TABLE]

and using estimate (3.8) of property $(b3),$ we have that the second term is

[TABLE]

for all $n$ large. The inequality (3.20) follows by setting $x=p_{n}$ in the estimate $-\log(1-x)>x$ (see (3.2)). The final estimate (3.21) follows from the first estimate (3.5) of property $(b1).$

Using estimates (3.21) and (3.19) in the expression for $A_{1}$ in (3.15), we have

[TABLE]

for all $n$ large. The final estimate follows using $1-p_{n}<1.$ We now fix $\gamma$ as in the statement of the Theorem and choose $\epsilon=\epsilon(\gamma,\eta)>0$ small so that

[TABLE]

for all $n$ large. This proves $(c1).$

We prove $(c2)$ as follows. Using the upper bound for $L_{n}\log{n}$ in (3.19) and the lower bound for $v_{L_{n}}$ in property $(b3),$ we have

[TABLE]

where the final estimate (3.22) follows from the fact that $\delta_{n}=\epsilon p_{n}$ and the lower bound for $p_{n}$ in (3.5) (see property $(b1)$). As before, we choose $\epsilon=\epsilon(\eta,\gamma)>0$ small so that the final term in (3.22) is at least $n^{2\eta-2\gamma-\alpha_{1}}$ for all $n$ large. This proves (3.16).

Proof of $(ii)$

Fix $\eta$ and $\gamma$ as in the statement of the Theorem. Fix $\epsilon>0$ small to be determined later and let $M=M(\epsilon)\geq 2$ be large so that

[TABLE]

Set $q_{n}=n,$

[TABLE]

and

[TABLE]

Here $\epsilon_{1}=\epsilon_{1}(\epsilon)>0$ is to be determined later. For a fixed $\epsilon>0,$ we have the following estimates regarding $L_{n}$ and $\delta_{n}.$

$(b1)$ We have that

[TABLE]

and

[TABLE]

for all $n$ large.

$(b2)$ We have

[TABLE]

for all $n$ large. If $\epsilon_{1}>0$ is sufficiently small, then

[TABLE]

for all $n$ large.

$(b3)$ There is a constant $N_{0}=N_{0}(\eta,\epsilon)\geq 1$ such that

[TABLE]

for all $n$ large.

Proof of $(b1)-(b3)$: The property $(b1)$ is true as follows. Since $\alpha_{1}=0,$ we have from (1.5) that $\log\left(\frac{1}{p_{n}}\right)\leq\epsilon\log{n}=\log\left(n^{\epsilon}\right)$ for all $n$ large. This proves the first inequality in (3.26). Since $\alpha_{2}=0,$ we have from (1.6) that $\log\left(\frac{1}{1-p_{n}}\right)\leq\epsilon\log{n}=\log\left(n^{\epsilon}\right)$ for all $n$ large. This proves the second inequality of (3.26).

To prove (3.27), we proceed as follows. If $\delta_{n}=\epsilon_{1}p_{n},$ then we have from the first inequality in (3.26) that $\delta_{n}\geq\epsilon_{1}n^{-\epsilon}\geq n^{-2\epsilon}$ for all $n$ large. If $\delta_{n}=\epsilon(1-p_{n}),$ then using the second inequality in (3.26), we have $\delta_{n}\geq\epsilon n^{-\epsilon}\geq n^{-2\epsilon}$ for all $n$ large.

The first estimate (3.28) follows by using the lower bound $-\log(1-x)>x$ with $x=1-p_{n}$ in the definition of $L_{n}$ in (3.24). The second estimate in (3.28) follows using $\eta<1.$ The final estimate follows from (3.26). To prove (3.29), we consider two cases separately depending on whether $\delta_{n}=\epsilon p_{n}$ or $\delta_{n}=(1-\epsilon)p_{n}.$ If $\delta_{n}=\epsilon_{1}p_{n},$ then $p_{n}<1-\frac{1}{M}$ and so we have

[TABLE]

if $\epsilon_{1}=\epsilon_{1}(\epsilon)>0$ is small.

If $\delta_{n}=\epsilon(1-p_{n}),$ then $p_{n}\geq 1-\frac{1}{M}$ and $(1+\epsilon)(1-p_{n})\leq\frac{1+\epsilon}{M}<1$ since $M\geq 2$ and $0<\epsilon<1.$ Therefore using the upper bound estimate $-\log(1-x)<\frac{x}{1-x}$ from (3.1) with $x=(1+\epsilon)(1-p_{n}),$ we have

[TABLE]

Similarly using the lower bound estimate $-\log(1-x)>x$ from (3.2), we have

[TABLE]

Using the above two estimates, we have

[TABLE]

by our choice of $M$ from (3.23). This proves $(b2).$

To prove property $(b3),$ we argue as follows. Setting $q_{n}=n,L=L_{n}$ (as in (3.24)) in the definition of $v_{i}$ in (2.21) we have

[TABLE]

where

[TABLE]

for all $n$ large. The estimate in (3.32) follows since $\alpha_{2}$ defined in (1.5) is zero and so $\log\left(\frac{1}{1-p_{n}}\right)<\epsilon\log{n}$ for all $n$ large. The estimate in (3.33) follows from (3.29) and the definition of $L_{n}$ in (3.24).

Substituting (3.35) into (3.31), we get

[TABLE]

for all $n$ large. The second inequality follows using $1-p_{n}<1.$ This proves $(b3).$

We use properties $(b1)-(b3)$ to prove $(ii)$ in Theorem 1.

Proof of $(ii)$: We set $q=q_{n}=n$ and $L=L_{n}$ and $\delta=\delta_{n}$ as in (3.24) and (3.25), respectively, in the expressions for $A_{1}$ and $A_{2}$ in (2.23) and (2.24). We then have

[TABLE]

and

[TABLE]

The following estimates for $A_{1}$ and $A_{2}$ imply the lower bound (1.10) for case $(ii)$ in Theorem 1. Fix $\gamma,\eta>0$ as in the statement of the Theorem.

$(c1)$ There are positive constants $\epsilon=\epsilon(\gamma,\eta)>0$ and $M_{1}=M_{1}(\gamma,\eta,\alpha_{1},\epsilon)\geq 1$ so that

[TABLE]

for all $n\geq M_{1}.$

$(c2)$ There are positive constants $\epsilon=\epsilon(\gamma,\eta)>0$ and $M_{2}=M_{2}(\gamma,\eta,\alpha_{1},\epsilon)\geq 1$ so that

[TABLE]

for all $n\geq M_{2}.$

Proof of $(c1)-(c2)$: We first prove $(c1).$ Using the estimate (3.28) of property $(b2),$ we have that the first term in (3.37) is

[TABLE]

and using estimate (3.30) of property $(b3),$ we have that the second term in (3.37) is

[TABLE]

for all $n$ large. The inequality (3.42) follows using $-\log(1-x)>x$ for $0<x<1$ (see (3.2)). The final estimate in (3.43) follows from the first estimate (3.26) of property $(b1).$

Using estimates (3.43) and (3.41) in (3.37), we have

[TABLE]

for all $n$ large provided $\epsilon=\epsilon(\eta,\gamma)>0$ is small. This proves $(c1).$

We prove $(c2)$ as follows. Using the upper bound for $L_{n}$ in property $(b2)$ and the lower bound for $v_{L_{n}}$ in property $(b3),$ we have

[TABLE]

for all $n$ large, provided $\epsilon=\epsilon(\eta,\gamma)>0$ is small. The estimate (3) follows from the estimate for $\delta_{n}$ in (3.27). This gives the estimate $(c2)$ for the term $A_{2}.$

Proof of $(iii)$

Fix $\eta$ and $\gamma$ as in the statement of the Theorem. Fix $\epsilon>0$ small to be determined later and let $\alpha_{2}=\alpha_{2}>0$ be as defined in (1.6). Set $q_{n}=n,$

[TABLE]

and

[TABLE]

For a fixed $\epsilon>0,$ we have the following estimates regarding $L_{n}$ and $\delta_{n}.$

$(b1)$ We have that $\alpha_{1}=0$ and

[TABLE]

and

[TABLE]

for all $n$ large.

$(b2)$ We have

[TABLE]

and

[TABLE]

for all $n$ large.

$(b3)$ We have that

[TABLE]

for all $n$ large.

Proof of $(b1)-(b3)$: The property $(b1)$ is true as follows. From the definition of $\alpha_{2}>0$ in (1.6) we have that

[TABLE]

for all $n$ large. This proves the first inequality in (3.48). The second inequality follows from the first inequality since

[TABLE]

for all $n$ large. This also proves that $\alpha_{1}$ defined in (1.5) is zero. This proves the estimate (3.48) of property $(b1).$ To prove (3.49), we use (3.48) and obtain

[TABLE]

for all $n$ large. This proves $(b1).$

The first estimate (3.50) follows by using the lower bound $-\log(1-x)>x$ with $x=1-p_{n}$ in the definition of $L_{n}$ in (3.24). The second estimate in (3.28) follows using $1-\eta-\alpha_{2}<1.$ The final estimate follows from (3.48). The proof of (3.51) is analogous as the proof of (3.29) for the case $\delta_{n}=\epsilon(1-p_{n}).$ This proves $(b2).$

To prove property $(b3),$ we argue as follows. Setting $q_{n}=n,L=L_{n}$ (as in (3.46)) in the definition of $v_{i}$ in (2.21) we have

[TABLE]

where

[TABLE]

The final equality is true using $\delta_{n}=\epsilon(1-p_{n}).$ For the middle term, we use the lower bound for $1-p_{n}$ from (3.48) to get

[TABLE]

We evaluate the first term in (3.55) as follows. Since $\alpha_{2}<1,$ we have using (3.51) and the definition of $L_{n}$ in (3.46) that

[TABLE]

Substituting (3.57) and (3.56) into (3.55), we have

[TABLE]

for all $n$ large. Substituting (3.58) into (3.54), we get

[TABLE]

for all $n$ large. The final inequality follows from the estimate (3.48) in property $(b1).$ This proves $(b3)$ for the case $\alpha_{2}<1.$

We use properties $(b1)-(b3)$ to prove $(iii)$ in Theorem 1.

Proof of $(iii)$: We set $q=q_{n}=n$ and $L=L_{n}$ and $\delta=\delta_{n}$ as in (3.46) and (3.47), respectively, in the expressions for $A_{1}$ and $A_{2}$ in (2.23) and (2.24). We then have

[TABLE]

and

[TABLE]

The following estimates for $A_{1}$ and $A_{2}$ imply the lower bound (1.12) in case $(iii)$ of Theorem 1. Fix $\gamma,\eta>0$ as in the statement of the Theorem.

$(c1)$ There are positive constants $\epsilon=\epsilon(\gamma,\eta)>0$ and $M_{1}=M_{1}(\gamma,\eta,\alpha_{1},\epsilon)\geq 1$ so that

[TABLE]

for all $n\geq M_{1}.$

$(c2)$ There are positive constants $\epsilon=\epsilon(\gamma,\eta)>0$ and $M_{2}=M_{2}(\gamma,\eta,\alpha_{1},\epsilon)\geq 1$ so that

[TABLE]

for all $n\geq M_{2}.$

Proof of $(c1)-(c2)$: We first prove $(c1).$ Using the estimate (3.50) of property $(b2),$ we have that the first term in (3.60) is

[TABLE]

and using estimate (3.52) of property $(b3),$ we have that the second term in (3.60) is

[TABLE]

for all $n$ large. The inequality (3.65) follows using $-\log(1-x)>x$ for $0<x<1$ (see (3.2)). The final estimate in (3.66) follows from the final estimate (3.48) of property $(b1).$

Using estimates (3.66) and (3.64) in (3.60), we have

[TABLE]

for all $n$ large, provided $\epsilon=\epsilon(\eta,\gamma)>0$ is small. This proves $(c1).$

We prove $(c2)$ as follows. Using the upper bound for $L_{n}$ in property $(b2)$ and the lower bound for $v_{L_{n}}$ in property $(b3),$ we have

[TABLE]

for all $n$ large, provided $\epsilon=\epsilon(\eta,\gamma)>0$ is small. The estimate (3) follows from the estimate for $\delta_{n}$ in (3.49). This gives the estimate $(c2)$ for the term $A_{2}.$

4 Proof of Proposition 2 and Theorem 2

Proof of Proposition 2: By definition of $\alpha_{2}$ in (1.6), we have

[TABLE]

so that

[TABLE]

for all $n$ large. Since $\alpha_{2}>2,$ we fix $\epsilon>0$ small so that $\alpha_{2}-\epsilon>2.$ If $N_{e}$ denote the number of open edges in the random graph $G(n,p_{n}),$ we then have

[TABLE]

as $n\rightarrow\infty.$ But $\{N_{e}=0\}=\{\omega(G(n,p_{n}))=1\}$ and so we obtain (1.13).

An analogous proof holds for the other case $\alpha_{2}>2$ by considering closed edges.

If $1<\alpha_{2}<2,$ we argue as follows. If $W_{e}$ denotes the number of closed edges, then using the Binomial estimate (2.2), we have

[TABLE]

Using (4.1), we have

[TABLE]

and

[TABLE]

for all $n$ large. The first inequality in (4.5) is obtained using (4.1) and ${n\choose 2}\geq\frac{n^{2}}{4}$ for all $n$ large. We choose $\epsilon>0$ small so that

[TABLE]

We then have from (4.3), (4.5) and (4.4) that

[TABLE]

for all $n$ large. Suppose now that the event $W_{e}\leq(1+\epsilon)\frac{1}{2}n^{2-\alpha_{2}+\epsilon}$ occurs and let ${\cal S}_{e}$ be the set of all vertices belonging to the closed edges in the random graph $G(n,p_{n}).$ The induced subgraph $G_{S}$ with vertex set $\{1,2,\ldots,n\}\setminus{\cal S}_{e}$ contains at least $n-(1+\epsilon)n^{2-\alpha_{2}+\epsilon}$ vertices and every edge in $G_{S}$ is open. In other words, the graph $G_{S}$ is an open clique containing at least

[TABLE]

vertices, for all $n$ large.

We now prove the upper bound (1.17). For integer $q\geq 1,$ let $G(q,p)$ be the random graph with vertex set $S_{q}=\{1,2,\ldots,q\}.$ For integer $L\geq 2,$ let $B_{L}(S_{q})$ denote the event that the random graph $G(q,p)$ contains an open $L-$clique; i.e., there are vertices $\{v_{i}\}_{1\leq i\leq L}$ such that the edge between $v_{i}$ and $v_{j}$ is open for any $1\leq i\neq j\leq L.$ We have

[TABLE]

where

[TABLE]

We now set $q=n,p=p_{n}$ and let $f_{n}\rightarrow\infty$ be any sequence as in the statement of the Theorem. Setting $L=U_{n}$ as defined in (1.16), we then have $A_{0}=A_{0}(n)=f_{n}.$ This proves the upper bound (1.17) in Lemma 2.

Proof of $(i)$: Here $\alpha_{1}$ defined in (1.5) equals $\theta_{1}$ and $\alpha_{2}$ as defined in (1.6) equals zero. The lower bound follows from (1.8), case $(i)$ of Theorem 1. For the upper bound, we fix $\xi>0$ and set $f_{n}=\xi\log{n}$ so that $U_{n}$ as defined in (1.16) equals $\frac{2+\xi}{\theta_{1}}+1.$ The upper bound then follows from (1.17).

Proof of $(ii)$: Here $\alpha_{1}$ and $\alpha_{2}$ defined in (1.5) and (1.6), respectively, both equal zero. Fixing $\eta,\gamma$ as in the statement of $(ii),$ the lower bound follows from (1.10), case $(ii)$ of Theorem 1.

For the upper bound, we fix $0<\xi_{1}<\xi<1$ and set $f_{n}=\xi_{1}\log{n}.$ The term $U_{n}$ defined in (1.16) equals

[TABLE]

for all $n\geq N_{1}.$ Here $N_{1}=N_{1}(\xi,\xi_{1},p)\geq 1$ is a constant. Using (1.17) of Theorem 1, we have

[TABLE]

for all $n\geq N_{1}.$ Choosing $\xi_{1}$ sufficiently close to $\xi$ so that $\xi_{1}(2+\xi_{1})>\xi(1+\xi),$ we obtain the upper bound in $(ii).$

Proof of $(iii)$: Here $\alpha_{1}$ defined in (1.5) equals zero and $\alpha_{2}$ as defined in (1.6) equals $\theta_{2}.$ Let $\eta,\gamma>0$ be as in the statement of the Theorem and fix $\gamma_{0},\eta_{0}>0$ such that $\gamma_{0}-\frac{\theta_{2}}{2}<\gamma-\frac{\theta_{2}}{2}<\eta_{0}<\eta<1-\theta_{2}$ and $\eta_{0}-\gamma_{0}>\eta-\gamma.$ Let $\epsilon>0$ be small to be determined later. Applying the lower bound (1.12), case $(iii)$ in Theorem 1 with $\eta_{1}$ and $\gamma_{1}$ we have

[TABLE]

where the final estimate follows from the choices of $\eta_{0}$ and $\gamma_{0}.$ We have

[TABLE]

for all $n\geq N_{1}.$ Here $N_{1}=N_{1}(\epsilon)\geq 1$ is a constant. The first inequality in (4.10) follows from (3.1) and the second inequality follows from the fact that $1-p_{n}<\epsilon$ for all $n\geq N_{1}$ large. From (4.10), we therefore have

[TABLE]

provided $\epsilon=\epsilon(\eta_{1},\eta,\theta_{2})>0$ is small. Fixing such an $\epsilon$ and substituting the estimate (4.11) into (4.9), we obtain the lower bound in (1.20).

For the upper bound, we fix $\xi>0$ and set $f_{n}=\xi\log{n}.$ The term $U_{n}$ defined in (1.16) is then

[TABLE]

and using (3.2), we have

[TABLE]

Using the bounds (4.10) and (4.13) in (4.12), we have

[TABLE]

for all $n\geq N_{1}.$ Here $N_{1}=N_{1}(\epsilon)\geq 1$ is the constant in (4.10).

Using the above bounds in the upper bound (1.17) of Theorem 1, we have

[TABLE]

for all $n\geq N_{1}.$ Choosing $\epsilon>0$ small so that $(2+\xi)(1-\epsilon)>1+\xi,$ we obtain the upper bound in (1.20).

5 Proof of Theorem 3

For a graph $G=(V,E)$ on $n$ vertices, let $\alpha(G)$ be the independence number of the graph $G$ defined as as follows. For integer $0\leq h\leq n,$ we say that $\alpha(G)=h$ if and only if the following two conditions are satisfied:

$(a)$ There is a set of $h$ vertices, none of which have an edge between them.

$(b)$ If $h+1\leq n,$ then every set of $h+1$ vertices have an edge between them.

As in Section 1, let $\omega(G)$ denote the clique number of $G.$ Let $\overline{G}=(\overline{V},\overline{E})$ denote the compliment of the graph $G$ defined as follows. The vertex set $\overline{V}=V$ and an edge $e\in\overline{E}$ if and only if $e\notin E.$ The following three properties are used to prove Theorem 3.

$(d1)$ We have

[TABLE]

$(d2)$ We have

[TABLE]

$(d3)$ Suppose for some integer $1\leq m\leq n,$ every set of $m$ vertices in the complement graph $\overline{G}$ contains a clique of size $L.$ We then have

[TABLE]

The lower bounds in Theorem 3, follow from the respective upper

bounds (1.18), (1.19) and (1.20) on the clique number $\omega(G(n,1-r_{n}))$ of Theorem 2 and property $(d2)$ above. This is because, the random graph $\overline{G}(n,r_{n})$ has the same distribution as the random graph $G(n,1-r_{n}).$

For the upper bounds, we consider each case separately.

Proof of $(i)$: Here $r_{n}=\frac{1}{n^{\theta_{2}}}$ for some $\theta_{2}>0.$ To estimate the chromatic number using property $(d3),$ we identify cliques in subsets of the random graph $G(n,1-r_{n}).$ Fix $\beta>0$ to be determined later and set $m=n^{1-\beta}$ and apply Theorem 2, case $(iii)$ for the random graph $G(m,p_{m}),$ where

[TABLE]

where $\theta_{22}=\frac{\theta_{2}}{1-\beta}.$ We then have $\alpha_{1}=0$ and $\alpha_{2}=\theta_{22},$ where $\alpha_{1}$ and $\alpha_{2}$ are as defined in (1.5) and (1.6), respectively. Let $\eta,\gamma>0$ be such that

[TABLE]

From the proof of lower bound of (1.20), there is a positive integer $N_{3}=N_{3}(\eta,\gamma,\xi)\geq 1$ so that

[TABLE]

for all $m$ large, where

[TABLE]

The final estimate above follows using $m=n^{1-\beta}.$

Let ${\cal S}_{m}$ be the set of subsets of size $m$ in $\{1,2,\ldots,n\}$ and for a set $S\in{\cal S}_{m},$ let $F_{n}(S)$ denote the event that the random induced subgraph of $G(n,1-r_{n})$ with vertex set $S$ contains an open $L-$clique. From (5.6) we have that

[TABLE]

for all $n$ large. Let $F_{n}=\bigcap_{S\in{\cal S}_{m}}F_{n}(S)$ denote the event that every set of $m$ vertices in the random graph $G(n,1-r_{n})$ contains an $L-$clique. Since there are ${n\choose m}$ sets in ${\cal S}_{m},$ we have

[TABLE]

where

[TABLE]

The final estimate follows since $m=n^{1-\beta}.$ From the choices of $\eta$ and $\gamma$ in (5.5), we have that $2\eta-2\gamma+\theta_{22}>1$ and so

[TABLE]

for all $n\geq N_{2}.$ Here $N_{2}=N_{2}(\eta,\gamma,\beta)\geq 1$ is a constant and the final equality follows from the definition of $m=n^{1-\beta}$ and $\theta_{22}=\frac{\theta_{2}}{1-\beta}$ above.

If the event $F_{n}$ occurs, then using property $(d3),$ we have that

[TABLE]

Fixing $\beta>\theta_{2}$ and $\xi>0,$ we have that the final expression in (5.11) is at most

[TABLE]

for all $n$ large. Summarizing, we have from (5.9), (5.10) and (5.8) that

[TABLE]

We have the following property.

$(f1)$ Let

[TABLE]

Let $\xi,\zeta>0$ be as in the statement of the Theorem. There exists $(\eta,\gamma,\beta)\in{\cal T}$ such that

[TABLE]

and

[TABLE]

This proves the upper bound in (1.21) in Theorem 3.

Proof of $(f1)$: We recall that $\theta_{22}=\frac{\theta_{2}}{1-\beta}$ and we have the constraint that $\beta>\theta_{2}$ and $\gamma>0.$ Since

[TABLE]

we have from (5.5) that the least possible value for $\eta$ is $\frac{1-2\theta_{2}}{2(1-\theta_{2})};$ i.e., if $(\eta,\gamma,\beta)\in{\cal T},$ then $\eta\geq\frac{1-2\theta_{2}}{2(1-\theta_{2})}$ and $\beta>\theta_{2}.$

Fix $\delta>0$ small and fix $\theta_{2}<\beta<\theta_{2}+\delta$ and $\frac{1-2\theta_{2}}{2(1-\theta_{2})}<\eta<\frac{1-2\theta_{2}}{2(1-\theta_{2})}+\delta.$ From (5.15), we have that if $\delta,\gamma>0$ are sufficiently small, then $(\eta,\gamma,\beta)\in{\cal T}.$ Also, we have that $(1-\eta-\theta_{22})(1-\beta)\geq f(\theta_{2},\delta),$ where

[TABLE]

Since $f(\theta_{2},0)=\frac{1-2\theta_{2}}{2},$ we fix $\xi>0$ and choose $\delta=\delta(\xi)>0$ smaller if necessary so that

[TABLE]

This proves (5.13).

For (5.14), we proceed analogously and use the fact that $\eta\geq\frac{1-2\theta_{2}}{2(1-\theta_{2})}$ to obtain

[TABLE]

provided $\delta,\gamma>0$ are small.

Proof of $(ii)$: Here $r_{n}=p$ for some $p\in(0,1)$ and for all $n\geq 2.$ As in the proof of $(i)$ above, we identify cliques in subsets of the random graph $G(n,1-r_{n}).$ Fix $\beta>0$ to be determined later and set $m=n^{1-\beta},p_{m}=1-p$ and apply Theorem 2, case $(ii)$ for the random graph $G(m,1-p).$ We then have $\alpha_{1}=\alpha_{2}=0,$ where $\alpha_{1}$ and $\alpha_{2}$ are as defined in (1.5) and (1.6), respectively. Fixing

[TABLE]

we have from the proof of lower bound of (1.19), that there is a positive integer $N=N(\eta,\gamma)\geq 1$ so that

[TABLE]

for all $m\geq N_{3},$ where

[TABLE]

The final estimate above follows using $m=n^{1-\beta}.$

As in case $(i),$ let $F_{n}$ denote the event that every set of $m$ vertices in the random graph $G(n,1-r_{n})$ contains an $L-$clique. Analogous to (5.8), we then have

[TABLE]

where

[TABLE]

The final estimate follows since $m=n^{1-\beta}.$ From the choices of $\eta$ and $\gamma$ in (5.16), we have that $2\eta-2\gamma>1$ and so

[TABLE]

for all $n\geq N_{2}.$ Here $N_{2}=N_{2}(\eta,\gamma,\beta)\geq 1$ is a constant and the final equality follows from the definition of $m=n^{1-\beta}.$

If the event $F_{n}$ occurs, then using property $(d3),$ we have that

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Fixing $\beta>0$ and $\xi>0,$ we have that the final term in (5.11) is at most

[TABLE]

for all $n\geq N_{3}.$ Here $N_{3}=N_{3}(\eta,\beta,\xi)\geq 1$ is a constant. Summarizing, we have from (5.20), (5.21) and (5.19) that

[TABLE]

Analogous to property $(f1)$ above, we have the following property.

$(f2)$ Let

[TABLE]

and fix $\zeta>0.$ There exists $(\eta,\gamma,\beta)\in{\cal T}$ so that

[TABLE]

and

[TABLE]

Substituting the above into (5.23), we obtain the upper bound in (1.22) in Theorem 3.

Proof of $(f2)$: From (5.16), we have that the minimum possible value for $\eta$ is $\frac{1}{2}.$ Choosing $\eta>\frac{1}{2}$ sufficiently close to $\frac{1}{2}$ and $\beta>0$ small, both (5.24) and (5.25) are satisfied.

Proof of $(iii)$: Here $r_{n}=1-\frac{1}{n^{\theta_{1}}}$ for some $\theta_{1}>0.$ As before, we identify cliques in subsets of the random graph $G(n,1-r_{n}).$ Fix $\beta>0$ to be determined later and set $m=\beta n$ and apply Theorem 2, case $(iii)$ for the random graph $G(m,p_{m}),$ where

[TABLE]

We then have $\alpha_{1}=\theta_{1}$ and $\alpha_{2}=0,$ where $\alpha_{1}$ and $\alpha_{2}$ are as defined in (1.5) and (1.6), respectively. Let $\eta,\gamma>0$ be such that

[TABLE]

From the proof of lower bound of (1.18), we have that

[TABLE]

for all $m$ large, where

[TABLE]

As in cases $(i)-(ii),$ let $F_{n}$ denote the event that every set of $m$ vertices in the random graph $G(n,1-r_{n})$ contains an $L-$clique. Using ${n\choose k}\leq\left(\frac{ne}{k}\right)^{k}$ for integers $1\leq k\leq n,$ we have

[TABLE]

where

[TABLE]

The final estimate in (5.30) follows since $m=\beta n.$ From the choices of $\eta$ and $\gamma$ in (5.27), we have that $2\eta-2\gamma-\theta_{1}>1$ and so

[TABLE]

for all $n\geq N_{2}.$ Here $N_{2}=N_{2}(\eta,\gamma,\beta)\geq 1$ is a constant and the final equality follows from the definition of $m=\beta n.$

If the event $F_{n}$ occurs, then using property $(d3),$ we have that

[TABLE]

Summarizing, we have from (5.32) and (5.30) that

[TABLE]

for all $n$ large. We have the following property.

$(f3)$ Fix $\xi,\zeta>0$ and let

[TABLE]

There exists $(\eta,\gamma,\beta)\in{\cal T}$ such that

[TABLE]

and

[TABLE]

for all $n$ large. Using the above estimates in (5.33), we obtain the upper bound in (1.23) in Theorem 3.

Proof of $(f3)$: From (5.27), we have that the least possible value for $\eta$ is $\frac{1+\theta_{1}}{2}.$ Fix $\zeta,\xi>0.$ Choosing $\gamma,\beta>0$ sufficiently small and $\eta>\frac{1+\theta_{1}}{2}$ sufficiently close of $\frac{1+\theta_{1}}{2},$ we get that

[TABLE]

and $2\eta-2\gamma-\theta_{1}>1-\zeta.$ This proves (5.35) and (5.36).

Acknowledgement

I thank Professors Rahul Roy and Federico Camia for crucial comments and for my fellowships.