Solving Turán's tetrahedron problem for the ℓ₂-norm

Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron (Formula presented.). This is equivalent to determining the maximum (Formula presented.) -norm of the codegree vector of a (Formula presented.) -free (Formula presented.) -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, (Formula presented.), of a 3-uniform hypergraph (Formula presented.) is the sum of codegrees squared (Formula presented.) over all pairs of vertices (Formula presented.), or in other words, the square of the (Formula presented.) -norm of the codegree vector of the pairs of vertices. We define (Formula presented.) to be the maximum (Formula presented.) over all (Formula presented.) -free (Formula presented.) -vertex 3-uniform hypergraphs (Formula presented.). We use flag algebra computations to determine asymptotically the codegree squared extremal number for (Formula presented.) and (Formula presented.) and additionally prove stability results. In particular, we prove that the extremal (Formula presented.) -free hypergraphs in (Formula presented.) -norm have approximately the same structure as one of the conjectured extremal hypergraphs for Turán's conjecture. Further, we prove several general properties about (Formula presented.) including the existence of a scaled limit, blow-up invariance and a supersaturation result.