Random cliques in random graphs revisited

TL;DR

This work delivers a precise, two-way understanding of how copies of a fixed graph $H$ (notably $K_r$) are distributed inside the random graph $G(n,p)$, quantifying deviations from the independent-hypergraph model via a refined weighting by $p^{-t(H)}$ and a global correction $e^{-\Lambda(n,r)}$ that accounts for overlapping pairs (2-clusters). The authors develop a general probabilistic framework applicable to hypergraphs, derive a warm-up bound, then obtain a full upper bound (and two-sided bound) that isolates the effect of 2-clusters and bounds for larger clusters through cumulant-like quantities, culminating in concrete consequences for the counting of clique-factors near threshold. By instantiating the theory to cliques and clique-factors, the paper yields near-tight results for the distribution of $K_r$-copies in $G(n,p)$, provides two-sided bounds, and establishes upper bounds for the number of $K_r$-factors above the threshold, with broader applicability to hypergraphs and to regimes where $p$ is constant and $r$ grows with $n$. The methods hinge on a Warnke–Janson–style conditioning scheme, careful accounting of 2-clusters via $\Delta_2$, and a decomposition into typical outcomes, enabling precise control of overlap phenomena and leading to new counting bounds that complement existing coupling-based approaches.

Abstract

We study the distribution of the set of copies of some given graph $H$ in the random graph $G(n,p)$, focusing on the case when $H = K_r$. Our main results capture the 'leading term' in the difference between this distribution and the 'independent hypergraph model', where (in the case $H = K_r$) each copy is present independently with probability $π= p^{\binom{r}{2}}$. As a concrete application, we derive a new upper bound on the number of $K_r$-factors in $G(n,p)$ above the threshold for such factors to appear. We will prove our main results in a much more general setting, so that they also apply to random hypergraphs, and also (for example) to the case when $p$ is constant and $r = r(n) \sim 2\log_{1/p}(n)$.

Theorems & Definitions (90)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Remark 2.1
• Theorem 3.1
• proof : Proof of Theorem \ref{['th_wu']}
• Theorem 4.1
• Remark 4.2
• Lemma 4.3