On the impossibility of certain $({n^2+n+k}_{n+1})$ configurations

TL;DR

This paper proves the impossibility of certain combinatorial configurations based on divisibility and perfect square conditions, extending previous results and analyzing specific cases like $k=3$ using incidence matrices.

Contribution

It generalizes known impossibility results for $({n^2+n+k}_{n+1})$ configurations and provides new conditions for their existence based on modular arithmetic and perfect squares.

Findings

For $k=2$, conditions relate to parity and perfect squares.

For all $k$, configurations with transitive parallelism are constrained by modular conditions.

Specifically, the case $k=3$ is fully analyzed with new impossibility results.

Abstract

This paper investigates the impossibility of certain $({n^2+n+k}_{n+1})$ configurations. Firstly, for $k=2$, the result of \cite{gropp1992non} that $\frac{n^2+n}{2}$ is even and $n+1$ is a perfect square or $\frac{n^2+n}{2}$ is odd and $n-1$ is a perfect square is reproved using the incidence matrix $N$ and analysing the form of $N^TN$. Then, for all $k$, configurations where paralellism is a transitive property are considered. It is then analogously established that if $n\equiv0$ or $n\equiv k-1$ mod $k$ for $k$ even, then $\frac{n^2+n}{k}$ is even and $n+1$ is a perfect square or $\frac{n^2+n}{k}$ is odd and $n-(k-1)$ is a perfect square. Finally, the case $k=3$ is investigated in full generality.