Tur\'{a}n problems in pseudorandom graphs

TL;DR

This paper investigates the maximum density of pseudorandom graphs avoiding certain subgraphs, improving bounds for containing the Peterson graph and constructing dense triangle-free pseudorandom graphs, advancing understanding of extremal pseudorandom structures.

Contribution

It introduces an improved embedding procedure that tightens bounds on pseudorandom graphs avoiding specific subgraphs, and constructs new dense pseudorandom graphs with particular forbidden configurations.

Findings

Pseudorandom graphs with density > n^{-1/3} contain the Peterson graph.

Constructed densest known pseudorandom K_{2,3}-free and triangle-free graphs.

Provided a new proof for the absence of large cliques in certain pseudorandom graphs.

Abstract

Given a graph $F$, we consider the problem of determining the densest possible pseudorandom graph that contains no copy of $F$. We provide an embedding procedure that improves a general result of Conlon, Fox, and Zhao which gives an upper bound on the density. In particular, our result implies that optimally pseudorandom graphs with density greater than $n^{-1/3}$ must contain a copy of the Peterson graph, while the previous best result gives the bound $n^{-1/4}$. Moreover, we conjecture that the exponent $1/3$ in our bound is tight. We also construct the densest known pseudorandom $K_{2,3}$-free graphs that are also triangle-free. Finally, we obtain the densest known construction of clique-free pseudorandom graphs due to Bishnoi, Ihringer and Pepe in a novel way and give a different proof that they have no large clique.