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Tetrahedron Conjecture in the $\ell_2$-norm

Levente Bodnár, Wanfang Chen, Jinghua Deng, Jianfeng Hou, Xizhi Liu, Jialei Song, Jiabao Yang, Yixiao Zhang

TL;DR

The paper resolves the Turán-type question for the tetrahedron $K_{4}^{3}$ in the $ ext{L}_2$-norm by proving that the balanced cyclic 3-partite construction $oldsymbol{ C}_n$ uniquely maximizes the $ ext{L}_2$-norm for large $n$. The authors develop Mantel-type results for vertex-colored graphs, including both $L_1$ and $L_2$ norms and a stability theory within a Simonovits-style framework, with a computer-assisted flag-algebra component for the $L_2$-norm case. A two-phase local-modification approach increases the $ ext{L}_2$-norm while transforming near-extremal graphs toward the extremal construction, leading to a contradiction unless the graph is isomorphic to $oldsymbol{ C}_n$. The results advance our understanding of stability in higher-norm Turán problems and introduce techniques potentially applicable to a broader class of extremal questions.

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.

Tetrahedron Conjecture in the $\ell_2$-norm

TL;DR

The paper resolves the Turán-type question for the tetrahedron in the -norm by proving that the balanced cyclic 3-partite construction uniquely maximizes the -norm for large . The authors develop Mantel-type results for vertex-colored graphs, including both and norms and a stability theory within a Simonovits-style framework, with a computer-assisted flag-algebra component for the -norm case. A two-phase local-modification approach increases the -norm while transforming near-extremal graphs toward the extremal construction, leading to a contradiction unless the graph is isomorphic to . The results advance our understanding of stability in higher-norm Turán problems and introduce techniques potentially applicable to a broader class of extremal questions.

Abstract

The famous Tetrahedron Conjecture of Turán from the 1940s asserts that the number of edges in an -vertex -graph without the tetrahedron, the complete -graph on four vertices, cannot exceed that of the balanced complete cyclic -partite -graph, whose edges are of types , , , and . A recent surprising result of Balogh-Clemen-Lidický [J. Lond. Math. Soc. (2) 106 (2022)] shows that this conjecture is asymptotically true in the -norm, where the number of edges is replaced by the sum of squared codegrees. They further conjectured that, in this -norm setting, the -partite construction is uniquely extremal for large . 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 -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.

Paper Structure

This paper contains 10 sections, 14 theorems, 148 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mantel's theorem in vertex-colored graphs
  3. Proof of Theorem \ref{['THM:3-colored-Mantel-L1-norm']}
  4. Proof of Theorem \ref{['THM:3-colored-Mantel-stability']}
  5. Proof of Theorem \ref{['THM:3-colored-Mantel-L2-norm']}
  6. Proof of Theorem \ref{['THM:L2-exact-K43']}
  7. Preliminaries
  8. Two key lemmas
  9. Proof of Theorem \ref{['THM:L2-exact-K43']}
  10. Concluding remarks

Key Result

Theorem 1.1

We have $\mathrm{ex}_{\ell_{2}}(n, K_{4}^{3}) = \frac{n^4}{6} + o(n^4)$.

Figures (2)

  • Figure 1: Structures of $\mathbb{C}[V_1, V_2, V_3]$ and $\mathbb{B}[V_1, V_2]$.
  • Figure 2: Bad and missing edges in Lemmas \ref{['LEMMA:K43-L2-improvement-phase-one']} and \ref{['LEMMA:K43-L2-improvement-phase-two']}.

Theorems & Definitions (51)

  • Theorem 1.1: BCL22b
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.7
  • proof : Proof of Lemma \ref{['LEMMA:deg-sum-distinct-walk']}
  • proof : Proof of Theorem \ref{['THM:3-colored-Mantel-L1-norm-reduced']}
  • Claim 2.8
  • ...and 41 more