Random Polynomial Graphs for Random Tur\'an Problems

TL;DR

This paper extends bounds on Turán numbers from deterministic graphs to random graphs using random polynomial graphs, providing new lower bounds for Turán problems in probabilistic settings.

Contribution

It introduces effective lower bounds for Turán numbers in random graphs, generalizing previous deterministic results to the probabilistic context.

Findings

Effective lower bounds on $ ext{ex}(G_{n,p}, ext{T}^ ext{ell})$

Bounds for generalized Turán numbers in random graphs

Extension of deterministic bounds to probabilistic models

Abstract

Bukh and Conlon used random polynomial graphs to give effective lower bounds on $\mathrm{ex}(n,\mathcal{T}^\ell)$, where $\mathcal{T}^\ell$ is the $\ell$th power of a balanced rooted tree $T$. We extend their result to give effective lower bounds on $\mathrm{ex}(G_{n,p},\mathcal{T}^\ell)$, which is the maximum number of edges in a $\mathcal{T}^\ell$-free subgraph of the random graph $G_{n,p}$. Analogous bounds for generalized Tur\'an numbers in random graphs are also proven.