A random version of the Burr-Erd\H{o}s-Spencer theorem

TL;DR

This paper introduces a probabilistic variant of a classical theorem in graph Ramsey theory, extending the scope of previous results to random settings for collections of graphs.

Contribution

It generalizes the Burr-Erdős-Spencer theorem to a random context, broadening its applicability in probabilistic combinatorics.

Findings

Proves a random version of the Burr-Erdős-Spencer theorem.

Extends the random Ramsey theorem of R"odl and Ruciński.

Provides new insights into probabilistic graph coloring.

Abstract

A well-known result of Burr, Erd\H{o}s and Spencer [Transactions of the American Mathematical Society, 1975] determines the $2$-colour Ramsey number for any sufficiently large collection of vertex-disjoint copies of a fixed graph $H$ without isolated vertices. In this short note we prove a random version of this result, thereby generalising the random Ramsey theorem of R\"odl and Ruci\'nski [Journal of the American Mathematical Society, 1995].