# Books versus triangles at the extremal density

## TL;DR

This paper explores the relationship between the number of triangles and the maximum number of triangles sharing an edge (book number) in dense graphs, providing new bounds and confirming a conjecture for specific parameter ranges.

## Contribution

It proves Mubayi's conjecture about the dependency between triangle count and book number for certain ranges of the parameter 2, advancing understanding of extremal graph configurations.

## Key findings

- Confirmed Mubayi's conjecture for 2=1/6 and 24=0.2495 to 1/4.
- Established lower bounds on the number of triangles based on book number constraints.
- Connected extremal graph properties with triangle density in dense graphs.

## Abstract

A celebrated result of Mantel shows that every graph on $n$ vertices with $\lfloor n^2/4 \rfloor + 1$ edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must in fact be at least $\lfloor n/2 \rfloor$ triangles in any such graph. Another strengthening, due to the combined efforts of many authors starting with Erd\H{o}s, says that any such graph must have an edge which is contained in at least $n/6$ triangles. Following Mubayi, we study the interplay between these two results, that is, between the number of triangles in such graphs and their book number, the largest number of triangles sharing an edge. Among other results, Mubayi showed that for any $1/6 \leq \beta < 1/4$ there is $\gamma > 0$ such that any graph on $n$ vertices with at least $\lfloor n^2/4\rfloor + 1$ edges and book number at most $\beta n$ contains at least $(\gamma -o(1))n^3$ triangles. He also asked for a more precise estimate for $\gamma$ in terms of $\beta$. We make a conjecture about this dependency and prove this conjecture for $\beta = 1/6$ and for $0.2495 \leq \beta < 1/4$, thereby answering Mubayi's question in these ranges.