Hypergraph Lagrangians I: the Frankl-F\"uredi conjecture is false

TL;DR

This paper disproves the longstanding Frankl-F"uredi conjecture by providing counterexamples for all uniformities r ≥ 4, while also identifying conditions under which the conjecture holds for large parameters.

Contribution

It introduces an infinite family of counterexamples to the Frankl-F"uredi conjecture for r ≥ 4 and delineates the parameter range where the conjecture remains valid.

Findings

Counterexamples for all r ≥ 4

Conjecture holds for large t within specific bounds

Disproves the universality of the conjecture

Abstract

An old and well-known conjecture of Frankl and F\"{u}redi states that the Lagrangian of an $r$-uniform hypergraph with $m$ edges is maximised by an initial segment of colex. In this paper we disprove this conjecture by finding an infinite family of counterexamples for all $r \ge 4$. We also show that, for sufficiently large $t \in \mathbb{N}$, the conjecture is true in the range $\binom{t}{r} \le m \le \binom{t+1}{r} - \binom{t-1}{r-2}$.