A note on a problem of Erdos and Rothschild

TL;DR

This paper investigates the minimal size of the largest book in graphs with a given number of vertices and edges, where every edge is part of at least one triangle, providing bounds based on the graph's edge density.

Contribution

It establishes new lower bounds on the largest book size in graphs with specified edge counts and triangle coverage constraints.

Findings

Derived bounds depend on the function f(n) related to edge count

Showed that bk(G) grows at least as fast as min(n/√f(n), n²/f(n)²)

Provides insights into the structure of dense triangle-rich graphs

Abstract

A set of $q$ triangles sharing a common edge is a called a book of size $q$. Letting $bk(G)$ denote the size of the largest book in a graph $G$, Erd\H{o}s and Rothschild \cite{erdostwo} asked what the minimal value of $bk(G)$ is for graphs $G$ with $n$ vertices and a set number of edges where every edge is contained in at least one triangle. In this paper, we show that for any graph $G$ with $n$ vertices and $\frac{n^2}{4} - nf(n)$ edges where every edge is contained in at least one triangle, $bk(G) \geq \Omega\left(\min{\{\frac{n}{\sqrt{f(n)}}, \frac{n^2}{f(n)^2}\}}\right)$.