## Context

This is exercise 1.7.1 of Tao/Vu's Additive Combinatorics, generally considered a graduate level math textbook.

My question is not "How do I solve this exercise?"

My question is "What is this exercise even looking for?"

## Definition

We define a graph $G(n,p)$ as follows: for each pair of nodes, we independently decide, with probability $p=n^{-1+\epsilon}$, whether to assign edge between the pair of nodes.

Let $Y$ be the number of triangles in the graph.

## Problem:

Provide upper and lower bonds for $P(Y > 3/2 E(Y))$.

## What I have so far (the upper bound):

Proving $P(Y > 3/2 E(Y)) < ... $ is fairly easy. This is a matter of calculating $E(Y)$, $E_1(Y)$, $E_2(Y)$, ... and applying theorem 1.37 (bounds on correlation of polynomials.) To avoid ruining the exercise, more will not be said about the upper bound.

## What I'm stuck on (the lower bound):

I can't figure out what $...$ should look like in $P(Y > 3/2E(Y)) > ... $.

Everything I've seen in the book so far deals with proving statements of the form:

$P( | Y - E(Y) | > ... ) < ... $ -- it provides an upper bound on deviation from the mean.

Now, however, I want to prove a *lower* bound on deviation from the mean.

## Question:

What types of inequalities should I be looking into?

Is there some trivial way to "flip" the bounds that I'm not seeing?

Thanks!