On the number of $A$-transversals in hypergraphs

TL;DR

This paper investigates the maximum number of $A$-transversals in hypergraphs, providing sharp bounds for special cases like strongly independent sets, and extends classical results to more general transversal conditions.

Contribution

It introduces a general framework for counting $A$-transversals in hypergraphs and establishes sharp bounds for specific cases, generalizing known theorems like Moon-Moser.

Findings

Sharp bounds for the number of maximal strongly independent sets in 3-uniform hypergraphs.

Extension of classical theorems to $A$-transversals with various sets $A$.

Identification of conditions under which hypergraphs lack certain $A$-transversals.

Abstract

A set $S$ of vertices in a hypergraph is \textit{strongly independent} if every hyperedge shares at most one vertex with $S$. We prove a sharp result for the number of maximal strongly independent sets in a $3$-uniform hypergraph analogous to the Moon-Moser theorem. Given an $r$-uniform hypergraph ${\mathcal H}$ and a non-empty set $A$ of non-negative integers, we say that a set $S$ is an \textit{$A$-transversal} of ${\mathcal H}$ if for any hyperedge $H$ of ${\mathcal H}$, we have \mbox{$|H\cap S| \in A$}. Independent sets are $\{0,1,\dots,r{-}1\}$-transversals, while strongly independent sets are $\{0,1\}$-transversals. Note that for some sets $A$, there may exist hypergraphs without any $A$-transversals. We study the maximum number of $A$-transversals for every $A$, but we focus on the more natural sets, e.g., $A=\{a\}$, $A=\{0,1,\dots,a\}$ or $A$ being the set of odd or the set of even numbers.