On asymmetric hypergraphs

TL;DR

This paper proves the existence of infinitely many minimal asymmetric hypergraphs for any uniformity $k \\ge 3$ and determines the minimum size of such hypergraphs for all $k \\ge 1$, highlighting a stark contrast with the $k=2$ case.

Contribution

It establishes the existence of infinitely many minimal asymmetric $k$-uniform hypergraphs for all $k \\ge 3$ and finds the minimum size for asymmetric hypergraphs across all $k \\ge 1$.

Findings

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

Minimum size of asymmetric hypergraphs determined for all $k \\ge 1$.

Contrast with the finite case for $k=2$, where only 18 minimal asymmetric graphs exist.

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.