Tur\'an densities of hypercubes

TL;DR

This paper advances the semidefinite flag algebra method to improve upper bounds on Turán densities for hypercubes and hypergraphs, providing new bounds and methodological enhancements.

Contribution

It introduces an improved semidefinite flag algebra approach applicable to hypercubes and hypergraphs, yielding tighter upper bounds on Turán densities.

Findings

Improved upper bound of 0.60318 for edge Turán density of 4-cycle free subcube

New upper bound of 0.36577 for forbidding 6-cycles in hypercubes

Vertex Turán density of Q_3 is at most 0.76900

Abstract

In this paper we describe a number of extensions to Razborov's semidefinite flag algebra method. We will begin by showing how to apply the method to significantly improve the upper bounds of edge and vertex Tur\'an density type results for hypercubes. We will then introduce an improvement to the method which can be applied in a more general setting, notably to 3-uniform hypergraphs, to get a new upper bound of 0.5615 for $\pi(K_4^3)$. For hypercubes we improve Thomason and Wagner's result on the upper bound of the edge Tur\'an density of a 4-cycle free subcube to 0.60318 and Chung's result on forbidding 6-cycles to 0.36577. We also show that the upper bound of the vertex Tur\'an density of $\mc{Q}_3$ can be improved to 0.76900, and that the vertex Tur\'an density of $\mc{Q}_3$ with one vertex removed is precisely 2/3.