Tetrahedron Conjecture in the $\ell_2$-norm

TL;DR

This paper confirms the conjecture that the balanced 3-partite construction uniquely maximizes the sum of squared codegrees in a 3-graph without tetrahedra under the ll_2-norm, extending Ture1n's classical problem.

Contribution

It proves the uniqueness of the extremal 3-partite construction in the ll_2-norm setting for the Tetrahedron Conjecture, using novel stability methods and a Mantel theorem for vertex-colored graphs.

Findings

Confirmed the conjecture of unique extremality of the 3-partite construction.

Developed a Mantel theorem for vertex-colored graphs with forbidden triangles.

Introduced a new procedure in Simonovits' stability method applicable to extremal problems.

Abstract

The famous Tetrahedron Conjecture of Tur\'an from the 1940s asserts that the number of edges in an $n$-vertex $3$-graph without the tetrahedron, the complete $3$-graph on four vertices, cannot exceed that of the balanced complete cyclic $3$-partite $3$-graph, whose edges are of types $V_1 V_2 V_3$, $V_1 V_1 V_2$, $V_2 V_2 V_3$, and $V_3 V_3 V_1$. A recent surprising result of Balogh-Clemen-Lidick\'y [J. Lond. Math. Soc. (2) 106 (2022)] shows that this conjecture is asymptotically true in the $\ell_2$-norm, where the number of edges is replaced by the sum of squared codegrees. They further conjectured that, in this $\ell_2$-norm setting, the $3$-partite construction is uniquely extremal for large $n$. We confirm this conjecture. Two key ingredients in our proofs include establishing a Mantel theorem for vertex-colored graphs that forbid certain types of triangles, and introducing a novel procedure integrated into Simonovits' stability method, which essentially reduces the task to verifying that the $\ell_2$-norm of certain near-extremal constructions increases under suitable local modifications. The strategy in the latter may be of independent interest and potentially applicable to other extremal problems.