The maximum number of triangles in $K_{1,s,t}$-free graphs

TL;DR

This paper investigates the maximum number of triangles in graphs that do not contain a specific bipartite subgraph, providing a new elementary proof that improves the upper bound without relying on the triangle removal lemma.

Contribution

It offers a new elementary proof for the upper bound on the number of triangles in $K_{1,s,t}$-free graphs, improving previous bounds and avoiding complex tools.

Findings

Improved upper bound: $O(n^{3-1/s}(rac{1}{ ext{log } n})^{1-1/s})$

Elementary proof avoiding triangle removal lemma

Enhanced understanding of triangle counts in bipartite-free graphs

Abstract

We consider the following generalized Tur\'an problem: For $2 \le s \le t$, what is the maximum number of triangles in a $K_{1,s,t}$-free graph on $n$ vertices? The previously best known lower and upper bounds are $\Omega(n^2)$ and $o(n^{3-1/s})$, respectively. To the best of our knowledge, all known proofs of the upper bound use the triangle removal lemma. We give a new elementary proof that avoids the use of the triangle removal lemma and improves the upper bound to $O\left(n^{3-1/s}(\log n)^{-1+1/s}\right)$.