Solving Turán's tetrahedron problem for the ℓ2$\ell _2$‐norm

Turán's famous tetrahedron problem is to compute the Turán density of K 4 3 $K_4^3$ . This equivalent determining maximum ℓ 1 $\ell _1$ -norm codegree vector a -free n $n$ -vertex 3-uniform hypergraph. We introduce new way for measuring extremality hypergraphs and determine asymptotically extremal function in our notion. The squared sum, co 2 ( G ) $\mbox{co}_2(G)$ , hypergraph $G$ sum codegrees d x y $d(x,y)^2$ over all pairs vertices $xy$ or other words, square _2$ vertices. define exco H $\mbox{exco}_2(n,H)$ be $H$ use flag algebra computations number 5 $K_5^3$ additionally prove stability results. In particular, we that have approximately same structure as one conjectured conjecture. Further, several general properties about including existence scaled limit, blow-up invariance supersaturation result.