Solving Turán's Tetrahedron Problem for the ${\ell }_2$-Norm

Abstract

Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron $K_4^3$. This is equivalent to determining the maximum ${\ell }_1$-norm of the codegree vector of a $K_4^3$-free $n$-vertex $3$-uniform hypergraph. We introduce a new way for measuring extremality of hypergraphs and determine asymptotically the extremal function of the tetrahedron in our notion. The codegree squared sum, ${\text{co}}_2(G)$, of a $3$-uniform hypergraph $G$ is the sum of codegrees squared $d(x,y{)}^2$ over all pairs of vertices $xy$, or in other words, the square of the ${\ell }_2$-norm of the codegree vector of the pairs of vertices. We define ${\text{exco}}_2(n,H)$ to be the maximum ${\text{co}}_2(G)$ over all $H$-free $n$-vertex $3$-uniform hypergraphs $G$. We use flag algebra computations to determine asymptotically the codegree squared extremal number for $K_4^3$ and $K_5^3$ and additionally prove stability results. In particular, we prove that the extremal $K_4^3$-free hypergraphs in ${\ell }_2$-norm have approximately the same structure as one of the conjectured extremal hypergraphs for Turán's conjecture. Further, we prove several general properties about ${\text{exco}}_2(n,H)$ including the existence of a scaled limit, blow-up invariance and a supersaturation result.

arXiv comment23 pages

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