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

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 ${\mathrm{\ell }}_2$-norm, where the number of edges is replaced by the sum of squared codegrees. They further conjectured that, in this ${\mathrm{\ell }}_2$-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 ${\mathrm{\ell }}_2$-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.