Sabotage the Mantel Theorem

TL;DR

This paper explores how the maximum edges in triangle-free graphs are affected when a specific subgraph is required, providing bounds that are tight for certain random graph models.

Contribution

It introduces bounds on the maximum edges in triangle-free graphs containing a prescribed subgraph, extending Mantel's theorem to more constrained graph classes.

Findings

Bounds are tight for random triangle-free graphs.

Bounds are tight for graphs from the triangle-free process.

Provides general upper and lower bounds for the problem.

Abstract

One of the earliest results in extremal graph theory, Mantel's theorem, states that the maximum number of edges in a triangle-free graph $G$ on $n$ vertices is $\lfloor n^2/4 \rfloor$. We investigate how this extremal bound is affected when $G$ is additionally required to contain a prescribed graph $\mathbb{P}$ as a subgraph. We establish general upper and lower bounds for this problem, which are tight in the exponent for random triangle-free graphs and graphs generated by the triangle-free process, when the size of $\mathbb{P}$ lies within certain ranges.