On Turán numbers of the complete 4-graphs

Abstract

The Turán number T(n,α+1,r) is the minimum number of edges in an n-vertex r-graph whose independence number does not exceed α. For each r≥2, there exists t⁎(r) such that T(n,α+1,r)=t⁎(r)nrα1−r(1+o(1)) as α/r→∞ and n/α→∞. It is known that t⁎(2)=1/2, and the conjectured value of t⁎(3) is 2/3. We prove that t⁎(4)<0.706335.