# Maximizing 2-Independent Sets in 3-Uniform Hypergraphs

## TL;DR

This paper determines the hypergraph structure that maximizes the number of 2-independent sets in 3-uniform hypergraphs with a given number of edges, extending to broader uniform hypergraph cases.

## Contribution

It solves the problem of identifying the hypergraph with the maximum 2-independent sets for fixed edges, introducing a lex-style hypergraph as optimal, and discusses generalizations.

## Key findings

- A (2,3,1)-lex style 3-graph is optimal for maximizing 2-independent sets.
- The problem relates to maximizing independent sets in graphs with a fixed number of triangles.
- Proposes conjectures for asymptotic solutions in broader hypergraph settings.

## Abstract

There has been interest recently in maximizing the number of independent sets in graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. Similarly, it is a corollary of the Kruskal-Katona theorem that the lex graph has the maximum number of independent sets in a graph of fixed size and order. In this paper we solve two equivalent problems. The first is: what $3$-uniform hypergraph on a ground set of size $n$, having at least $t$ edges, has the most $2$-independent sets? Here a $2$--independent set is a subset of vertices containing fewer than $2$ vertices from each edge. This is equivalent to the problem of determining which graph on $n$ vertices having at least $t$ triangles has the most independent sets. The (hypergraph) answer is that, ignoring some transient and some persistent exceptions, a $(2,3,1)$-lex style $3$-graph is optimal. We also discuss the problem of maximizing the number of $s$-independent sets in $r$-uniform hypergraphs of fixed size and order, proving some simple results, and conjecture an asymptotically correct general solution to the problem.