I am interested in the following problem I tried thinking about and was unable to find a reliable answer to yet. What is the best concentration result on the amount of complete subgraphs of $G_{n,1/2}$ of size $\log n$?
Is there some theorem that states something similar to the following: Let $X$ be a random variable, denoting the number of cliques of size $\log n$ in a graph generated according to $G_{n,1/2}$ model, is it true that: $Pr(|X-EX| > \lambda EX) < e^{-c\lambda^2}$ for say some constant $c$ which is not a function of $n$? Note that I am not too picky on the formulation, and the distance from the mean can be taken in terms of standard deviations, also the right hand side doesn't matter to me too much as long as its subexponential.
I know that there are a lot of theorems regarding the destribution of a fixed subgraph $H$ of $G_{n,p}$, the best one I have found was a paper by Van H. Vu but these results are not relevant to my question as my query is about the distribution of a cliques of logarithmic size, which are by definition not a fixed subgraph.