On Hypergraph Lagrangians and Frankl-Füredi's Conjecture

Hui Lei; Linyuan Lu

paper

On Hypergraph Lagrangians and Frankl-Füredi's Conjecture

arXiv:1806.11259

Abstract

Frankl and Füredi conjectured in 1989 that the maximum Lagrangian, denoted by $λ_{r} (m)$ , among all $r$ -uniform hypergraphs of fixed size $m$ is achieved by the minimum hypergraph $C_{r, m}$ under the colexicographic order. We say $m$ in {\em principal domain} if there exists an integer $t$ such that $(r t - 1) \leq m \leq (r t) - (r - 2 t - 2)$ . If $m$ is in the principal domain, then Frankl-Füredi's conjecture has a very simple expression: $λ_{r} (m) = \frac{1}{( t - 1 ) ^{r}} (r t - 1) .$ Many previous results are focusing on $r = 3$ . For $r \geq 4$ , Tyomkyn in 2017 proved that Frankl-Füredi's conjecture holds whenever $(r t - 1) \leq m \leq (r t) - (r - 2 t - 2) - δ_{r} t^{r - 2}$ for a constant $δ_{r} > 0$ . In this paper, we improve Tyomkyn's result by showing Frankl-Füredi's conjecture holds whenever $(r t - 1) \leq m \leq (r t) - (r - 2 t - 2) - δ_{r}^{'} t^{r - \frac{7}{3}}$ for a constant $δ_{r}^{'} > 0$ .

16 pages