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

Abstract

Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron K43$K_4^3$. This is equivalent to determining the maximum ℓ1$\ell _1$‐norm of the codegree vector of a K43$K_4^3$‐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, co2(G)$\mbox{co}_2(G)$, of a 3‐uniform hypergraph G$G$ is the sum of codegrees squared dfalse(x,yfalse)2$d(x,y)^2$ over all pairs of vertices xy$xy$, or in other words, the square of the ℓ2$\ell _2$‐norm of the codegree vector of the pairs of vertices. We define exco2(n,H)$\mbox{exco}_2(n,H)$ to be the maximum co2(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 K43$K_4^3$ and K53$K_5^3$ and additionally prove stability results. In particular, we prove that the extremal K43$K_4^3$‐free hypergraphs in ℓ2$\ell _2$‐norm have approximately the same structure as one of the conjectured extremal hypergraphs for Turán's conjecture. Further, we prove several general properties about exco2(n,H)$\mbox{exco}_2(n,H)$ including the existence of a scaled limit, blow‐up invariance and a supersaturation result.

Hypergraph Turán Problems in $\ell_2$-Norm

“…We [3] proved general properties of σ including the existency of the limit in (1). In Table 1 we present bounds and exact values for the codegree squared density of various hypergraphs.…”

“…holds, where U (3) denotes the set of all three element subsets of U . The uniform Turán density π u (H) of a 3-graph H is defined to be π u (H) : = sup{d ∈ [0, 1] : for every η > 0 and n ∈ N there exists an H-free (d, η, 1)-dense hypergraph G with |V (G)| ≥ n}.…”

“…where (G n ) n≥1 is a convergent sequence with G n being an n-vertex 3-graph. For a well-written explanation of the flag algebra method in the setting of 3-uniform hypergraphs see [27] and for the particular application and an explanation of (2) see [3].…”

“…For hypergraphs, determining the Turán density is notoriously difficult; very few exact results are known. For example, the Turán density of the innocent looking tetrahedron K 3 4 , the complete 3-uniform hypergraph on 4 vertices, is unknown.…”

“…In order to get a better understanding of these problems, various different kinds of extremality such as the generalized Turán function or the minimum codegree threshold have been studied. We [3] recently introduced a new type of extremality for hypergraphs and solved the tetrahedron problem asymptotically for this notion. Here, we will systematically study extremal problems regarding this function.…”