Modular and Local Limit Theorems for Subgraph Counts

Definitions and mathematical statement

For an integer $n\geq 1$ and $p\in[0,1]$, the standard binomial random graph $G(n,p)$ has vertex set $\{1,\ldots,n\}$, with each unordered pair of distinct vertices included as an edge independently with probability $p$. The source does not specify whether $p$ is fixed or may depend on $n$, or which range of $p$ is intended.

Let $H$ be a fixed graph, independent of $n$, and let $X_H(n,p)$ denote the number of copies of $H$ in $G(n,p)$. This notation is provisional: the supplied record does not specify the graph class, whether copies are induced, or whether copies are counted as embeddings or as unlabelled subgraphs. These choices must be resolved before $X_H(n,p)$ is a uniquely defined statistic.

Source question. For every fixed graph $H$, do analogous modular-equidistribution and local-limit theorems hold for $X_H(n,p)$?

Unresolved qualifications. The word "analogous" refers to preceding results that are not included in the supplied record. A precise modular-equidistribution assertion requires a modulus or a range of moduli, the residue classes under consideration, a limiting regime, and an error bound or convergence criterion. A precise local-limit assertion requires a normalization, a candidate limiting density, the relevant lattice and its span, and the range over which probability estimates are to hold. None of these is specified here. Degenerate choices of $H$ or $p$, and any resulting restrictions on attainable counts or residues, also require explicit treatment. Thus the supplied question is preserved as a research target, but it does not determine a unique fully quantified conjecture.

Short English statement

For each fixed graph $H$, ask whether its copy count in $G(n,p)$ satisfies suitable modular-equidistribution and local-limit theorems. The supplied record leaves the comparison theorems, counting convention, parameter ranges, and limiting assertions unspecified, so a fully precise conjecture requires further source context.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001249$, source problem code catalogue entry, title Modular and Local Limit Theorems for Subgraph Counts; Oberwolfach Reports Open Problems collection, proposed year 2009. No individual proposer is identified.
• Supplied report citation: Combinatorics and Probability (2009), Oberwolfach Reports, DOI 10.4171/owr/2009/23, pp. 1225--1302. The record describes these reports as documenting mathematical workshops and their open-problem discussions. https://doi.org/10.4171/owr/2009/23
• Supplied literature source: "Sah-Sawhney local limit theorems." https://arxiv.org/abs/2006.11369
• Supplied literature source: "joint subgraph-count local limits." https://arxiv.org/abs/2412.09535
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports that later local-limit work proves broad analogues for connected subgraph counts and related joint distributions, while the conjectured full modular statement for every fixed $H$ is still not a single general theorem. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, explaining that this denotes meaning-preserving semantic, typographic, and LaTeX cleanup with no unsupported source-dependent content added. This supplied label does not resolve the formulation gaps identified above.