Minimal asymmetric hypergraphs

TL;DR

This paper proves the existence of infinitely many minimal asymmetric hypergraphs for any uniformity $k \,\ge\, 3$, contrasting with the finite case for graphs ($k=2$), and determines the minimal size of such hypergraphs.

Contribution

It establishes the existence of infinitely many minimal asymmetric hypergraphs for all uniformities $k \ge 3$ and finds the minimum size of asymmetric hypergraphs for all $k$.

Findings

Infinitely many minimal asymmetric hypergraphs exist for all $k \ge 3$.

For $k=2$, only 18 minimal asymmetric graphs exist.

Minimum size of asymmetric hypergraphs is determined for all $k$.

Abstract

In this paper, we prove that for any $k\ge 3$, there exist infinitely many minimal asymmetric $k$-uniform hypergraphs. This is in a striking contrast to $k=2$, where it has been proved recently that there are exactly $18$ minimal asymmetric graphs. We also determine, for every $k\ge 1$, the minimum size of an asymmetric $k$-uniform hypergraph.