Prague dimension of random graphs

TL;DR

This paper determines that the Prague dimension of binomial random graphs is typically of order n/log n for constant edge probabilities, using advanced hypergraph coloring techniques.

Contribution

It proves a conjecture about the typical Prague dimension of random graphs and introduces a new hypergraph edge-coloring result for large uniformities.

Findings

Prague dimension of binomial random graphs is of order n/log n

Established a new edge-coloring theorem for large uniformity hypergraphs

Confirmed conjecture by Furedi and Kantor regarding random graphs

Abstract

The Prague dimension of graphs was introduced by Nesetril, Pultr and Rodl in the 1970s. Proving a conjecture of Furedi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order n/log n for constant edge-probabilities. The main new proof ingredient is a Pippenger-Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size O(log n).