Conditional central limit conjecture for general subgraph counts in exponential random graphs

Let $H$) be a fixed graph containing $t$ triangles and $s$ two stars. Under the same conditions as in the two-star conditional central limit theorem, let $F_n$ denote the number of copies of $H$ in $G_n$. Define

$$(n)_{(k)}=n(n-1)\cdots(n-k+1),$$

let $\operatorname{Aut}(K)$ denote the number of automorphisms of $K$, and set

\mu_F=\frac{(n)_{(v(H))}}{\operatorname{Aut}(H)}p^{e(H)}+\frac{(n-3)_{(v(H)-3)}}{\operatorname{Aut}(H)}\left(2s p^{e(H)-2}\mu_{ \begin{tikzpicture}[scale=0.5] \draw (0,0) -- (0,0.4); \draw (-0.25,0) -- (0,0.4); \draw (0.25,0) -- (0,0.4); \end{tikzpicture} V}+6t p^{e(H)-3}\mu_{ \begin{tikzpicture}[scale=0.5] \draw (0,0) -- (0,0.4); \draw (-0.25,0) -- (0,0.4); \draw (0.25,0) -- (0,0.4); \end{tikzpicture} \triangle}\right), \sigma_F^2=\left(\frac{(n-3)_{(v(H)-3)}}{\operatorname{Aut}(H)}\right)^2\left(4s^2p^{2e(H)-4}\sigma_V^2+36t^2p^{2e(H)-6}\sigma_{ \begin{tikzpicture}[scale=0.5] \draw (0,0) -- (0,0.4); \draw (-0.25,0) -- (0,0.4); \draw (0.25,0) -- (0,0.4); \end{tikzpicture} \triangle}^2\right),

where \begin{tikzpicture}[scale=0.5] \draw (0,0) -- (0,0.4); \draw (-0.25,0) -- (0,0.4); \draw (0.25,0) -- (0,0.4); \end{tikzpicture} \triangle=\sum_{i<j<k} \begin{tikzpicture}[scale=0.5] \draw (0,0) -- (0,0.4); \draw (-0.25,0) -- (0,0.4); \draw (0.25,0) -- (0,0.4); \end{tikzpicture} Y_{ik} \begin{tikzpicture}[scale=0.5] \draw (0,0) -- (0,0.4); \draw (-0.25,0) -- (0,0.4); \draw (0.25,0) -- (0,0.4); \end{tikzpicture} Y_{jk} \begin{tikzpicture}[scale=0.5] \draw (0,0) -- (0,0.4); \draw (-0.25,0) -- (0,0.4); \draw (0.25,0) -- (0,0.4); \end{tikzpicture} Y_{ij}, and \mu_{ \begin{tikzpicture}[scale=0.5] \draw (0,0) -- (0,0.4); \draw (-0.25,0) -- (0,0.4); \draw (0.25,0) -- (0,0.4); \end{tikzpicture} V}, $\sigma_V$, \mu_{ \begin{tikzpicture}[scale=0.5] \draw (0,0) -- (0,0.4); \draw (-0.25,0) -- (0,0.4); \draw (0.25,0) -- (0,0.4); \end{tikzpicture} \triangle}, and \sigma_{ \begin{tikzpicture}[scale=0.5] \draw (0,0) -- (0,0.4); \draw (-0.25,0) -- (0,0.4); \draw (0.25,0) -- (0,0.4); \end{tikzpicture} \triangle} are the corresponding means and standard deviations defined in the paper. General subgraph-count conjecture. The normalized count satisfies

$$\frac{F_n-\mu_F}{\sigma_F}\longrightarrow N(0,1)$$

in distribution. This conjecture extends the conditional central limit theorem proved for two-star counts to general fixed subgraphs containing triangles and two stars; the paper gives only preliminary computations for this general case, so the claim remains open.