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Let $N$ be the number of copies of a small subgraph $H$ in anErd\H{o}s-R\'enyi graph $G \sim \mathcal{G}(n, p_n)$ where $p_n \to 0$ ischosen so that $\mathbb{E} N = c$, a constant. Results of Bollob\'as show thatfor regular graphs $H$, the count $N$ weakly converges to a Poisson randomvariable. For large but finite $n$, and for the specific case of the triangle,investigations of the upper tail $\mathbb{P}(N \geq k_n)$ by Ganguly, Hiesmayrand Nam (2022) revealed that there is a phase transition in the tail behaviorand the associated mechanism. Smaller values of $k_n$ correspond to disjointoccurrences of $H$, leading to Poisson tails, with a different behavioremerging when $k_n$ is large, guided by the appearance of an almost clique. Weshow that a similar phase transition also occurs when $H$ is any regular graph,at the point where $k_n^{1 -2/q}\log k_n = \log n$ ($q$ is the number ofvertices in $H$). This establishes universality of this transition, previouslyknown only for the case of the triangle.

Universality in prelimiting tail behavior for regular subgraph counts in the Poisson regime