# Singer difference sets and the projective norm graph

## TL;DR

This paper explores the relationship between Singer difference sets and projective norm graphs, providing new descriptions and proving the presence of specific complete bipartite subgraphs, which impacts extremal graph theory bounds.

## Contribution

It introduces a novel polynomial-based description of Singer difference sets within norm groups and proves the existence of a $K_{4,6}$ subgraph in projective norm graphs for all prime powers $q \

## Key findings

- Singer difference sets can be described as solutions to a polynomial equation.
- The projective norm graph $ ext{NG}(q,4)$ contains $K_{4,6}$ for all $q \
- The results extend the understanding of subgraph containment in norm graphs and their extremal properties.

## Abstract

We demonstrate a close connection between the classic planar Singer difference sets and certain norm equation systems arising from projective norm graphs. This, on the one hand leads to a novel description of planar Singer difference sets as a subset $\mathcal{H}$ of $\mathcal{N}$, the group of elements of norm 1 in the field extension $\mathbb{F}_{q^3}/\mathbb{F}_q$. $\mathcal{H}$ is given as the solution set of a simple polynomial equation, and we obtain an explicit formula expressing each non-identity element of $\mathcal{N}$ as a product $B\cdot C^{-1}$ with $B, C\in \mathcal{H}$. The description and the definitions naturally carry over to the nonplanar and the infinite setting. On the other hand, relying heavily on the difference set properties, we also complete the proof that the projective norm graph $\text{NG}(q,4)$ does contain the complete bipartite graph $K_{4,6}$ for every prime power $q \geq 5$. This complements the property, known for more than two decades, that projective norm graphs do not contain $K_{4,7}$ (and hence provide tight lower bounds for the Tur\'an number $ex(n,K_{4,7})$).