A Note on Generalized Lagrangians of Non-uniform Hypergraphs

Set A⊂ℕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A\subset {\mathbb N}$\end{document} is less than B⊂ℕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B\subset {\mathbb N}$\end{document} in the colex ordering if max(A△B)∈B. In 1980’s, Frankl and Füredi conjectured that the r-uniform graph with m edges consisting of the first m sets of ℕ(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb N}^{(r)}$\end{document} in the colex ordering has the largest Lagrangian among all r-uniform graphs with m edges. A result of Motzkin and Straus implies that this conjecture is true for r=2. This conjecture seems to be challenging even for r=3. For a hypergraph H=(V,E), the set T(H)={|e|:e∈E} is called the edge type of H. In this paper, we study non-uniform hypergraphs and define L(H) a generalized Lagrangian of a non-uniform hypergraph H in which edges of different types have different weights. We study the following two questions: 1. Let H be a hypergraph with m edges and edge type T. Let Cm,T denote the hypergraph with edge type T and m edges formed by taking the first m sets with cardinality in T in the colex ordering. Does L(H)≤L(Cm,T) hold? If T={r}, then this question is the question by Frankl and Füredi. 2. Given a hypergraph H, find a minimum subhypergraph G of H such that L(G) = L(H). A result of Motzkin and Straus gave a complete answer to both questions if H is a graph. In this paper, we give a complete answer to both questions for {1,2}-hypergraphs. Regarding the first question, we give a result for {1,r1,r2,…,rl}-hypergraph. We also show the connection between the generalized Lagrangian of {1,r1,r2,⋯ ,rl}-hypergraphs and {r1,r2,⋯ ,rl}-hypergraphs concerning the second question.