# A note on the maximum number of triangles in a $C_5$-free graph

## TL;DR

This paper establishes an upper bound on the maximum number of triangles in a $C_5$-free graph, improving previous estimates and contributing to extremal graph theory by refining triangle count limits.

## Contribution

The paper provides a tighter upper bound on the number of triangles in $C_5$-free graphs, advancing understanding in extremal combinatorics.

## Key findings

- Maximum number of triangles is at most (1/2√2)(1+o(1)) n^{3/2}
- Improves previous estimates by Alon and Shikhelman
- Refines bounds in extremal graph theory

## Abstract

We prove that the maximum number of triangles in a $C_5$-free graph on $n$ vertices is at most $\frac{1}{2 \sqrt 2} (1 + o(1)) n^{3/2}$, improving an estimate of Alon and Shikhelman.