Solving Turán's tetrahedron problem for the ℓ2$\ell _2$‐norm
József Balogh, Felix Christian Clemen, Bernard Lidický
Abstract
Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron K 4 3 $K_4^3$ . This is equivalent to determining the maximum ℓ 1 $\ell _1$ -norm of the codegree vector of a K 4 3 $K_4^3$ -free n $n$ -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, co 2 ( G ) $\mbox{co}_2(G)$ , of a 3-uniform hypergraph G $G$ is the sum of codegrees squared d ( x , y ) 2 $d(x,y)^2$ over all pairs of vertices x y $xy$ , or in other words, the square of the ℓ 2 $\ell _2$ -norm of the codegree vector of the pairs of vertices. We define exco 2 ( n , H ) $\mbox{exco}_2(n,H)$ to be the maximum co 2 ( G ) $\mbox{co}_2(G)$ over all H $H$ -free n $n$ -vertex 3-uniform hypergraphs G $G$ . We use flag algebra computations to determine asymptotically the codegree squared extremal number for K 4 3 $K_4^3$ and K 5 3 $K_5^3$ and additionally prove stability results. In particular, we prove that the extremal K 4 3 $K_4^3$ -free hypergraphs in ℓ 2 $\ell _2$ -norm have approximately the same structure as one of the conjectured extremal hypergraphs for Turán's conjecture. Further, we prove several general properties about exco 2 ( n , H ) $\mbox{exco}_2(n,H)$ including the existence of a scaled limit, blow-up invariance and a supersaturation result.