An improved construction for the triangle removal lemma

TL;DR

This paper presents an improved construction related to the triangle removal lemma, reducing the number of triangles in certain graphs while maintaining a high edge removal threshold, using additive combinatorics techniques.

Contribution

It introduces a new graph construction that improves the bounds on the triangle removal lemma, halving the previous constant and employing additive combinatorics methods.

Findings

Constructed graphs with fewer triangles for given edge removal

Achieved a new constant C_new ≈ 1.6601 in bounds

Used additive combinatorics, especially the corners problem

Abstract

We construct $n$-vertex graphs $G$ where $\epsilon n^2$ edges must be deleted to become triangle-free, which contain less than $\epsilon^{(C_{\text{new}}-o(1))\log_2 1/\epsilon}n^3$ triangles for $C_{\text{new}}= \frac{1}{4\log_2(4/3)} \approx 1.6601$. Previously, a bound of the same shape was known, but with $C_{\text{new}}$ replaced by $C_{\text{old}} := C_{\text{new}}/2$. Our construction uses ideas from additive combinatorics, drawing especially from the corners problem, but does not yield new bounds for those problems.