Triangle-free graph criterion for discontinuity in lower-tail probabilities

Let $H$ be a fixed graph, let $p\in(0,1)$ be constant, and set

$$m=p\binom{n}{2}.$$

Let $G_m\sim G(n,m)$, let $N_H(G_m)$ denote the number of copies of $H$ in $G_m$, and write

$$\mu_H=\mathbb{E}\left[N_H(G_m)\right].$$

Triangle-free graph discontinuity conjecture. The function

$$\beta\mapsto\frac{\log\log\mathbb{P}\left(N_H(G_m)<(1-n^\beta)\mu_H\right)^{-1}}{\log n}$$

has a jump discontinuity at $\beta=-1$ if and only if $H$ is triangle-free.

This conjecture proposes a general criterion for whether lower-tail moderate deviations exhibit a discontinuity near $\delta=n^{-1}$. The paper describes the problem as wide open beyond the triangle case and relates the bipartite-graph behavior to Siderenko's conjecture.