Random Polynomial Graphs for Random Turán Problems

TL;DR

This work extends the random polynomial graph method to Turán-type problems in random graphs and to generalized Turán numbers. It introduces a rooted-tree framework with density parameters, and develops a transference principle via local isomorphisms that moves lower bounds from fixed graphs to random-host settings. The main results give a.a.s. lower bounds of the form $\Omega\left(p^{1-\frac{a}{b}}n^{2-\frac{a}{b}}\right)$ for $\mathrm{ex}(G_{n,p},\mathcal{T}^\ell)$ when $\rho(T)\ge \frac{b}{a}$ and $pn\log n\to\infty$, and extend to generalized Turán numbers with bounds $\Omega\left(p^{e-\frac{a}{b}e}n^{v-\frac{a}{b}e}\right)$. The approach blends random-polynomial constructions, local isomorphism techniques, and algebraic-geometry tools to obtain effective lower bounds in random and relative Turán problems, with potential extensions to hypergraphs and broader combinatorial settings.

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 Turán numbers in random graphs are also proven.

Theorems & Definitions (25)

• Theorem 1.1: bukh2018rational
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• Definition 1
• Proposition 2.2
• Proposition 2.3
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm:mainGeneralized']}