A Large Deviation Result On The Number Of Small Subgraphs Of A Random Graph

Fix a small graph H and let YH denote the number of copies of H in the random graph G(n, p). We investigate the degree of concentration of YH around its mean, motivated by the following questions • For what λ does YH have sub-Gaussian behaviour, namely Pr(|YH − E(YH)| ≥ (λvar(YH))) ≤ e−cλ. where c is a positive constant. • What is Pr(YH ≥ (1 + 2)E(YH)), for a constant 2, i.e, when the upper tail is of order Ω(E(YH)) ? • Fixing λ = ω(1) in advance, find a reasonably small tail T = T (λ) such that Pr(|YH − E(YH)| ≥ T ) ≤ e−λ. We prove a general concentration result which contains a partial answer to each of these questions. §