Dense subgraphs in the H-free process

TL;DR

This paper analyzes the properties of the H-free process, showing that with high probability, the resulting graph lacks certain dense subgraphs of limited size, extending previous results for specific cases.

Contribution

It generalizes existing results by demonstrating that for strictly 2-balanced graphs H, the H-free process avoids dense subgraphs of bounded size, broadening understanding of graph evolution.

Findings

Final graphs contain no subgraphs with high density and limited vertices

Results hold with high probability as n approaches infinity

Extends previous work from C_3-free to general H-free processes

Abstract

The H-free process starts with the empty graph on n vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of H is created, where H is some fixed graph. When H is strictly 2-balanced, we show that for some c,d>0, with high probability as $n \to \infty$, the final graph of the H-free process contains no subgraphs F on $v_F \leq n^{d}$ vertices with maximum density $\max_{J \subseteq F}\{e_J/v_J\} \geq c$. This extends and generalizes results of Gerke and Makai for the C_3-free process.