Largest subgraph from a hereditary property in a random graph

TL;DR

This paper determines the maximum size of subgraphs within a random graph that belong to a hereditary family, linking it to the chromatic number of graphs outside the family.

Contribution

It establishes a probabilistic bound on the maximum edges in subgraphs from hereditary properties in random graphs, connecting it to chromatic numbers.

Findings

Maximum edges in subgraphs are asymptotically determined by chromatic number.

Results hold with high probability for large graphs.

Provides a probabilistic extension of classical extremal graph theory.

Abstract

We prove that for every non-trivial hereditary family of graphs ${\cal P}$ and for every fixed $p \in (0,1)$, the maximum possible number of edges in a subgraph of the random graph $G(n,p)$ which belongs to ${\cal P}$ is, with high probability, $$ \left(1-\frac{1}{k-1}+o(1)\right)p{n \choose 2}, $$ where $k$ is the minimum chromatic number of a graph that does not belong to ${\cal P}$.