Tetrahedron Conjecture in the ${\ell }_2$-norm

Abstract

The famous Tetrahedron Conjecture of Turán 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-Lidický [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.

arXiv commentproof of Fact 3.1 added, fixed/added some references