The largest $K_r$-free set of vertices in a random graph

TL;DR

This paper studies the maximum size of $K_r$-free subgraphs in random graphs, revealing a non-monotonic, interval-based concentration phenomenon that generalizes known results for independence numbers.

Contribution

It extends the understanding of $K_r$-free subgraph sizes in random graphs, showing complex non-monotonic behavior and generalizing the classical independence number results.

Findings

$ ext{alpha}_r(G)$ concentrates in a constant interval

Behavior varies non-monotonically with $n$

Results extend to other color critical graphs like $C_5$

Abstract

For $r \ge 2$ and a graph $G$, let $\alpha_{{r}}(G)$ be the maximum number of vertices in a $K_r$-free subgraph of $G$. We investigate the value $\alpha_{r}(G)$ when $G$ is the random graph $G \sim G_{n, 1/2}$ and discover the following phenomenon: with high probability, $\alpha_r(G)$ lies in an interval of constant length that varies in a non-monotonic fashion from $1$ to $\lfloor r/2\rfloor+1$ depending on the value of $n$. The special case $r=2$ corresponds to the independence number of random graphs which is well-known to have two-point concentration; our results therefore extend and generalize this basic fact in random graph theory, showing more complicated behavior when $r>2$. We also prove similar results where $K_r$ is replaced by any color critical graph like $C_5$.