On the Prague dimension of sparse random graphs

The Prague dimension of a graph $G$ is defined as the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph. Introduced in the 1970s by Nešetřil, Pultr, and Rödl – and motivated by the work of Dushnik and Miller, as well as by the induced Ramsey theorem – determining the Prague dimension of a graph is a notoriously hard problem. In this paper, we show that for all $\varepsilon > 0$ and $p$ such that $ n^{-1+\varepsilon} \le p \le n^{-\varepsilon}$, with high probability the Prague dimension of $G_{n,p}$ is $Θ_{\varepsilon}(pn)$, which improves upon a recent result by Molnar, Rödl, Sales and Schacht. Inspired by the work of Bennett and Bohman, our approach centres on analysing a random greedy process that builds an independent set of size $Ω(p^{-1}\log pn)$ by iteratively selecting vertices uniformly at random from the common non-neighbourhood of those already chosen. Using the differential equation method, we show that every non-edge is essentially equally likely to be covered by this process, which is key to establishing our bound.

💡 Research Summary

The paper “On the Prague dimension of sparse random graphs” addresses a fundamental and notoriously difficult problem in combinatorial graph theory: determining the Prague dimension of random graphs. The Prague dimension of a graph $G$ is defined as the minimum number of complete graphs (cliques) whose direct product contains $G$ as an induced subgraph. This parameter, rooted in the historical work of Nešetřil, Pultr, and Rödl, and connected to the Dushnik-Miller theorem and the induced Ramsey theorem, serves as a measure of the structural complexity of a graph.

The central contribution of this research is the establishment of a tight bound for the Prague dimension of the Erdős-Rényi random graph $G_{n,p}$ within a specific sparse regime. Specifically, for any $\varepsilon > 0$ and edge probability $p$ such that $n^{-1+\varepsilon} \le p \le n^{-\varepsilon}$, the authors prove that, with high probability, the Prague dimension of $G_{n,p}$ is $\Theta_{\varepsilon}(pn)$. This result represents a significant advancement over previous findings, particularly the work by Molnar, Rödl, Sales, and Schacht, by providing a more precise asymptotic characterization.

To achieve this, the authors employ a sophisticated probabilistic approach inspired by the random greedy process techniques developed by Bennett and Bohman. The methodology revolves around constructing an independent set of size $\Omega(p^{-1}\log pn)$ through an iterative, stochastic process. In this process, vertices are selected uniformly at random from the common non-neighborhood of the vertices already included in the set. This strategy is designed to systematically “cover” the non-edges of the graph, which is essential for bounding the Prague dimension.

The mathematical backbone of the paper is the application of the differential equation method. This technique allows the authors to model the discrete, stochastic evolution of the greedy process using continuous differential equations. By doing so, they are able to rigorously demonstrate that every non-edge in the graph is essentially equally likely to be covered by the evolving process. This property of “uniform coverage” is the critical technical breakthrough that enables the establishment of the $\Theta$ bound.

In summary, the paper provides a groundbreaking analysis of the Prague dimension in sparse random graphs. By combining a novel random greedy algorithm with the powerful differential equation method, the authors have successfully bridged the gap in our understanding of how the structural complexity of random graphs scales with the edge probability $p$. This work not only improves upon existing bounds but also introduces a robust framework for analyzing other complex graph parameters in the sparse regime.