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Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron K34. This is equivalent to determining the maximum ℓ1-norm of the codegree vector of a K34-free n-vertex 3-uniform hypergraph. We will 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, co2(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 ℓ2-norm of the codegree vector of the pairs of vertices. Define exco2(n,H) to be the maximum co2(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 K34 and K35 and additionally prove stability results. In particular, we prove that the extremal function for K34 in ℓ2-norm is asymptotically the same as the one obtained from one of the conjectured extremal K34-free hypergraphs for the ℓ1-norm. Further, we prove several general properties about exco2(n,H) including the existence of a scaled limit, blow-up invariance and a supersaturation result.
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Balogh, József; Clemen, Felix Christian; and Lidicky, Bernard, "Solving Turán's Tetrahedron Problem for the ℓ2-Norm" (2021). Mathematics Publications. 276.