# Lagrangians of Hypergraphs II: When colex is best

## TL;DR

This paper proves that colex segments maximize the Lagrangian for 3-uniform hypergraphs, provides a new proof for a related conjecture, and presents a counterexample to a previous conjecture, advancing hypergraph theory.

## Contribution

It confirms the Frankl-F"{u}redi conjecture for r=3, offers a new proof for Nikiforov's conjecture, and refutes an older conjecture of Ahlswede and Katona.

## Key findings

- Colex segments have the largest Lagrangian for 3-uniform hypergraphs.
- A new proof of Nikiforov's conjecture is provided.
- A counterexample disproves an old conjecture of Ahlswede and Katona.

## Abstract

A well-known conjecture of Frankl and F\"{u}redi from 1989 states that an initial segment of colex of has the largest Lagrangian of any $r$-uniform hypergraph with $m$ hyperedges. We show that this is true when $r=3$. We also give a new proof of a related conjecture of Nikiforov and a counterexample to an old conjecture of Ahlswede and Katona.