Some Motzkin-Straus type results for non-uniform hypergraphs

TL;DR

This paper extends Motzkin-Straus type results to non-uniform hypergraphs, linking maximum clique size and Lagrangian, with implications for Turán problems and extremal poset problems.

Contribution

It introduces new Motzkin-Straus type results specifically for non-uniform hypergraphs, expanding the theoretical framework beyond uniform cases.

Findings

Established Motzkin-Straus type inequalities for non-uniform hypergraphs

Connected hypergraph Lagrangian with Turán densities in non-uniform settings

Provided tools for extremal poset problem analysis

Abstract

A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in 1965. This connection and its extensions were applied in Tur\'{a}n problems of graphs and uniform hypergraphs. Very recently, the study of Tur\'{a}n densities of non-uniform hypergraphs has been motivated by extremal poset problems. In this paper, we give some Motzkin-Straus type results for non-uniform hypergraphs.