# Planar polynomials and an extremal problem of Fischer and Matousek

## TL;DR

This paper improves the lower bound on the maximum number of triangles in a 3-partite graph with no 4-cycles between parts, using affine planes and planar polynomials over finite fields.

## Contribution

It introduces a new construction based on planar polynomials that raises the lower bound for the maximum number of triangles in such graphs.

## Key findings

- Lower bound improved to (1 - o(1)) k^{5/3}
- Construction based on affine planes and planar polynomials
- Enhanced understanding of extremal configurations in tripartite graphs

## Abstract

Let $G$ be a 3-partite graph with $k$ vertices in each part and suppose that between any two parts, there is no cycle of length four. Fischer and Matou\u{s}ek asked for the maximum number of triangles in such a graph. A simple construction involving arbitrary projective planes shows that there is such a graph with $(1 - o(1)) k^{3/2} $ triangles, and a double counting argument shows that one cannot have more than $(1+o(1)) k^{7/4} $ triangles. Using affine planes defined by specific planar polynomials over finite fields, we improve the lower bound to $(1 - o(1)) k^{5/3}$.