Books vs Triangles

TL;DR

This paper explores the relationship between the size of the largest book in a graph and the minimum number of triangles it contains, establishing new asymptotic bounds and using advanced combinatorial theorems.

Contribution

It provides a new asymptotic bound linking book size constraints to triangle counts, combining and extending classical results in extremal graph theory.

Findings

For th, the number of triangles is at least th(1-th) n^2/4 - o(n^2).

Existence of eta > 0 such that for th, the graph contains at least eta n^3 triangles.

The results are asymptotically sharp and use the Ruzsa-Szemeredi theorem.

Abstract

A book of size b in a graph is an edge that lies in b triangles. Consider a graph G with n vertices and \lfloor n^2/4\rfloor +1 edges. Rademacher proved that G contains at least \lfloor n/2\rfloor triangles, and Erdos conjectured and Edwards proved that G contains a book of size at least n/6. We prove the following "linear combination" of these two results. Suppose that \alpha\in (1/2, 1) and the maximum size of a book in G is less than \alpha n/2. Then G contains at least \alpha(1-\alpha) n^2/4 - o(n^2) triangles as n approaches infinity. This is asymptotically sharp. On the other hand, for every \alpha\in (1/3, 1/2), there exists \beta>0 such that G contains at least \beta n^3 triangles. It remains an open problem to determine the largest possible \beta in terms of \alpha. Our proof uses the Ruzsa-Szemeredi theorem.