The lower tail for triangles in sparse random graphs

Let $X$ denote the number of triangles in the random graph $G(n,p)$. The problem of determining the asymptotic of the rate of the lower tail of $X$, that is, the function $f_c(n,p) = log Pr(X ≤ c E[X])$ for a given $c ∈ [0,1)$, has attracted considerable attention of both the combinatorics and the probability communities. We shall present a proof of the fact that whenever $p >> n^{-1/2}$, then $f_c(n,p)$ can be expressed as a solution to a natural combinatorial optimisation problem that generalises Mantel’s / Turan’s theorem. This is joint work with Gady Kozma.