Triangle-Count Equidistribution and Local Limits

Definitions and mathematical statement

Fix a real number $p\in(0,1)$, independent of $n$. Let $G_n\sim G(n,p)$ be the random simple graph on $\{1,\ldots,n\}$, with each unordered pair of distinct vertices independently present as an edge with probability $p$. Define its triangle count by \[ X_n=\sum_{1\le i<j<k\le n}\mathbf 1_{\{\{i,j\},\{i,k\},\{j,k\}\in E(G_n)\}}. \] Let $q=q(n)$ be a positive integer, and take all asymptotic limits as $n\to\infty$.

The supplied problem asks how far asymptotic equidistribution of $X_n$ modulo $q(n)$, and a local limit theorem for $X_n$, extend beyond the stated range \[ q(n)=o\!\left(\frac{n^2}{\log n}\right). \] The record describes this range as known, but does not supply the precise theorem or its error bounds. This description is retained as a source qualification, not independently verified here.

A precise target requires specifying the meaning of equidistribution when $q$ grows. For example, uniform relative error would mean \[ \max_{0\le a<q(n)}\left|q(n)\,\mathbb P\left(X_n\equiv a\pmod{q(n)}\right)-1\right|\longrightarrow0, \] whereas convergence in total variation to the uniform distribution would mean \[ \frac12\sum_{a=0}^{q(n)-1}\left|\mathbb P\left(X_n\equiv a\pmod{q(n)}\right)-\frac1{q(n)}\right|\longrightarrow0. \] These are distinct requirements for varying $q$; the supplied record does not select one.

Likewise, put $\mu_n=\mathbb E X_n$ and $\sigma_n^2=\operatorname{Var}(X_n)$. One possible uniform Gaussian local limit formulation, for $n\ge3$, is \[ \sup_{k\in\mathbb Z}\left|\sigma_n\,\mathbb P(X_n=k)-\frac1{\sqrt{2\pi}}\exp\!\left(-\frac{(k-\mu_n)^2}{2\sigma_n^2}\right)\right|\longrightarrow0. \] This is an illustrative interpretation, not an asserted theorem or a confirmed reconstruction of the source's intended local limit statement. The record does not specify the approximation, error scale, uniformity range in $k$, or how the modulus threshold is intended to constrain the local limit question.

Thus the source-supported research request is to determine the scope and limitations of extensions beyond the displayed modulus range, for every fixed $p\in(0,1)$, after resolving these formulation choices. The supplied literature discussion also concerns extensions from triangles to counts of fixed subgraphs, but does not specify a corresponding general-subgraph problem with sufficient hypotheses to replace this triangle-count question.

Short English statement

For a fixed edge probability $p\in(0,1)$, investigate the range of moduli for which the triangle count in $G(n,p)$ becomes asymptotically uniform, beyond the supplied threshold $q=o(n^2/\log n)$, and clarify the associated local limit question. The intended error criterion and the connection between the modulus range and the local limit theorem remain unspecified.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30001248$, title Triangle-Count Equidistribution and Local Limits, exact source problem code catalogue entry. The supplied proposed year is 2009; no individual proposer is identified.
• Combinatorics and Probability (2009), Oberwolfach Reports, Report 23, pp. 1225--1302, DOI 10.4171/owr/2009/23. Supplied primary report reference: https://doi.org/10.4171/owr/2009/23. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference labeled "DeMarco-Kahn-Redlich modular subgraph statistics": https://doi.org/10.37236/4094.
• Supplied literature reference labeled "Sah-Sawhney-Zhu joint local limit theorem": https://arxiv.org/abs/2412.09535.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially_solved and reports substantial positive results for extensions from triangle counts to fixed-subgraph counts, including modular equidistribution in sparse regimes and local limit theorems in broad connected-subgraph regimes, while saying that the full original analogue is not established. This is supplied dated curation, not independently verified. The background expressly calls for independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with a recorded qualification that the changes were meaning-preserving semantic, typographic, and LaTeX cleanup and added no unsupported source-dependent content. This metadata is not independent mathematical verification of the formulation or literature status.