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, blowup invariance and a supersaturation result.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.10408
 Bibcode:
 2021arXiv210810408B
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 23 pages