Most probably trangle-free graphs

TL;DR

This paper investigates the probability of a randomly sampled subgraph being triangle-free in graphs slightly exceeding Mantel's maximum edge count, providing exact results for the case with one extra edge.

Contribution

It introduces a probabilistic variant of the Erdős-Rademacher problem and determines the exact maximum probability for the case with one additional edge.

Findings

Derived the maximum probability for the triangle-free subgraph when the original graph has one more edge than Mantel's bound.

Proposed a probabilistic approach to a classical extremal graph theory problem.

Extended understanding of the structure of near-extremal triangle-free graphs.

Abstract

The celebrated Mantel's theorem states that any triangle-free graph on $n$ vertices contains at most $\left\lfloor n^2/4\right\rfloor$ edges. It is natural to ask how many triangles must exist in a graph with more than $\left\lfloor n^2/4\right\rfloor$ edges--a problem known as the Erd\H{o}s-Rademacher problem. In this paper, we propose a probabilistic variant of this classic problem. Specifically, given an $n$-vertex graph $G$ with $\left\lfloor n^2/4\right\rfloor+i$ ($i>0$) edges, we choose the edges of $G$ independently with probability $p$, and the resulting new graph is triangle-free with a certain probability. Our goal is to maximize this probability by choosing $G$ appropriately. For the case where $G$ has $ \left\lfloor n^2/4\right\rfloor +1$ edges, we determine the exact maximum probability.