Random Tur\'an Problems for Graphs with a Vertex Complete to One Part

TL;DR

This paper advances the understanding of the random Turán problem for bipartite graphs with a vertex complete to one part by establishing tight bounds and introducing improved upper bounds using advanced probabilistic techniques.

Contribution

It provides tight bounds for new classes of bipartite graphs with a vertex complete to one part and introduces superior general upper bounds for the problem.

Findings

Established tight bounds for multiple bipartite graphs with a vertex complete to one part.

Developed new general upper bounds that outperform previous bounds in many cases.

Utilized dependent random choice and hypergraph container methods in proofs.

Abstract

Given a graph $F$, the random Tur\'an problem asks to determine the maximum number of edges in an $F$-free subgraph of $G_{n,p}$. Prior to this work, the only bipartite graphs $F$ with known tight bounds included certain classes of complete bipartite graphs and theta graphs. We greatly expand upon these examples by proving tight bounds for a number of bipartite graphs which have a vertex complete to one part. We also prove new general upper bounds for this problem which in many cases do significantly better than the only previous known general upper bound due to Jiang and Longbrake. Our proofs utilize dependent random choice together with the recent technique of balanced vertex supersaturation in conjunction with hypergraph containers.