An extension of Motzkin-Straus Theorem to non-uniform hypergraphs and its applications

TL;DR

This paper extends the Motzkin-Straus theorem to non-uniform hypergraphs with edges of size one or two, and applies this to determine Turán densities, connecting classical graph theory results to hypergraph extremal problems.

Contribution

It introduces a new Lagrangian definition for non-uniform hypergraphs and extends key theorems to this broader class, advancing hypergraph extremal theory.

Findings

Extended Motzkin-Straus theorem to non-uniform hypergraphs

Extended Erdős-Stone-Simonovits theorem to non-uniform hypergraphs

Provided tools for analyzing Turán densities in hypergraphs

Abstract

In 1965, Motzkin and Straus established a remarkable connection between the order of a maximum clique and the Lagrangian of a graph and provided a new proof of Tur\'an's theorem using the connection. The connection of Lagrangians and Tur\'{a}n densities can be also used to prove the fundamental theorem of Erd\"{o}s-Stone-Simonovits on Tur\'{a}n densities of graphs. Very recently, the study of Tur\'{a}n densities of non-uniform hypergraphs have been motivated by extremal poset problems. In this paper, we attempt to explore the applications of Lagrangian method in determining Tur\'{a}n densities of non-uniform hypergraphs. We first give a definition of the Lagrangian of a non-uniform hypergraph, then give an extension of Motzkin-Straus theorem to non-uniform hypergraphs whose edges contain 1 or 2 vertices. Applying it, we give an extension of Erd\"{o}s-Stone-Simonovits theorem to non-uniform hypergraphs whose edges contain 1 or 2 vertices.