A Note on Lagrangians of \(4\)-Uniform Hypergraphs

Abstract

In 1989, Frankl and Füredi [1] conjectured that the \(r\)-uniform hypergraph with \(m\) edges formed by taking the first \(m\) sets in the colex ordering of \(\mathbb{N}^{(r)}\) has the largest Lagrangian of all \(r\)-uniform hypergraphs of size \(m\). For \(2\)-graphs, the Motzkin-Straus theorem implies this conjecture is true. For \(3\)-uniform hypergraphs, it was proved by Talbot in 2002 that the conjecture is true while \(m\) is in a certain range. In this paper, we prove that the \(4\)-uniform hypergraphs with \(m\) edges formed by taking the first \(m\) sets in the colex ordering of \(\mathbb{N}^{(r)}\) has the largest Lagrangian of all \(4\)-uniform hypergraphs with \(t\) vertices and \(m\) edges satisfying \(\binom{t-1}{4} \leq m \leq \binom{t-1}{4} + \binom{t-2}{3} – 17\binom{t-2}{2} + 1\).