On the Prague dimension of sparse random graphs

TL;DR

This paper determines the asymptotic Prague dimension of sparse Erdős–Rényi random graphs, showing it is proportional to pn for a wide range of p, using a novel probabilistic process analysis.

Contribution

It establishes a tight bound on the Prague dimension of sparse random graphs, improving previous results through a new probabilistic and differential equation approach.

Findings

Prague dimension of G_{n,p} is Θ_{ε}(pn) for specified p ranges.

Introduces a random greedy process to analyze graph properties.

Uses differential equations to show uniform coverage of non-edges.

Abstract

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 Ne\v{s}et\v{r}il, Pultr, and R\"odl -- 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 $\Theta_{\varepsilon}(pn)$, which improves upon a recent result by Molnar, R\"odl, 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 $\Omega(p^{-1}\log pn)$ 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.