Turán’s Tetrahedron Conjecture

• The topic presents Turán's (3,4)-conjecture, defining minimal edge conditions in 3-uniform hypergraphs with every 4-vertex set containing an edge.
• It introduces algebraic shifting and domination techniques that recast extremal hypergraph problems into an algebraic framework for verification.
• Methodological advances using flag algebras and semidefinite programming provide promising approaches for resolving longstanding open problems.

The Tetrahedron Conjecture of Turán, also referred to as Turán’s (3,4)-conjecture or the tetrahedron problem, occupies a central position in extremal hypergraph theory. It posits the minimal size or maximal density of a 3-uniform hypergraph on $n$ vertices in which every 4-vertex subset contains at least one edge. This conjecture has motivated developments across combinatorics, algebraic shifting, and modern applications of flag algebra methods. Although the (3,4)-conjecture remains open in its original form, significant progress has been made on its structural, algebraic, and computational aspects.

1. Combinatorial Formulation and Turán's Construction

Let $H$ be a 3-uniform hypergraph (3-graph) on $[n]$ such that every 4-set contains an edge. Turán's conjecture (1940) asserts that the minimum number of edges in such $H$ is achieved by a specific "balanced" 3-partite construction: partition $[n]$ into three parts $A_1, A_2, A_3$ (with sizes $s_1, s_2, s_3$ and $|s_i - s_j| \leq 1$), and include as edges all triples either contained within a single part or intersecting two vertices from one part and one from the next (cyclically). The conjectured minimum is: $$h(n) = \sum_{i=1}^3 \binom{s_i}{3} + \sum_{i=1}^3 s_i\binom{s_{i+1}}{2}$$ The combinatorial extremal configuration $C_n$ constructed this way is a balanced complete tripartite 3-graph with prescribed local edge patterns (Kalai et al., 2018).

2. Algebraic Shifting and Domination Reformulation

Algebraic shifting and domination provide an algebraic lens for recasting extremal problems. For Turán’s conjecture, the shifted family

$$C(n) = \{ \{a<b<c\} \subseteq [n] : a+c \leq n,\, 2a+b \leq n \}$$

satisfies $|C(n)| = h(n)$. The domination conjecture asserts that any 3-graph $H$ on $[n]$ with the "every 4-set spans an edge" property algebraically dominates $C(n)$, i.e., in the third compound matrix $C_3(X)$ of a generic $n \times n$ matrix $X$, the submatrix indexed by $C(n)$ and $H$ has full row rank. The exterior algebraic shift $H^{<_{c}}$, for a term order $<_{c}$ making $C(n)$ initial, satisfies $C(n) \subseteq H^{<_{c}}$ (Kalai et al., 2018).

Algebraic Shifting Definitions:

• For $k$-uniform families $F_1,F_2 \subseteq \binom{[n]}{k}$, $F_1$ dominates $F_2$ iff the corresponding compound matrix minor has maximal rank.
• The exterior algebraic shift $F^{<_{t}}$ constructs the lexicographically minimal weakly isomorphic shifted family.
• Combinatorial shifting applies elementary swaps, iterated until a shifted family is reached.

Algebraic shifting commutes with simplicial-complex skeletons, and key homological relations express Betti numbers in terms of shifted complexes.

3. Strengthened Conjectures and Graph Analogues

A strengthened version, termed the (3,4)-domination conjecture, postulates that all $H$ as above dominate $C(n)$—or, equivalently, $C(n) \subseteq H^{<_{c}}$. While open for hypergraphs, several graph analogues have been established:

• Domination version of Mantel-Turán: If $G$ is a graph on $[n]$ with every 3-set containing an edge, the edge family $E(G)$ dominates $B(n) = \{\{a<b\} : a+b \leq n\}$, the shifted extremal graph. Algebraically, $G^{<_{t}} \supseteq B(n)$ for any order refining the sum order $a+b$.
• Involution Theorem (graph analogue II): For a graph $G$ and involution $\tau$ on $[n]$, if every 3-set $S$ satisfies $|E(G[S])| + |E(G[\tau(S)])| \geq 2$, then $|E(G)| \geq |E(T(n))|$ where $T(n)$ is the balanced bipartite graph (Kalai et al., 2018).

The involution and algebraic domination results generalize and sharpen classical extremal graph theorems such as Mantel’s theorem.

4. Methodological Advances: Flag Algebras and Semidefinite Computation

Recent work introduces new analytic tools for the Turán tetrahedron problem, notably the flag algebra and semidefinite programming framework developed by Razborov. The extremality of a hypergraph is measured using the codegree vector—particularly its $\ell_1$ and $\ell_2$ norms:

• $\ell_1$-norm: Counts edges, corresponding to the classical problem.
• $\ell_2$-norm: The codegree-squared sum $$\mathrm{co}_2(G) = \sum_{\{x,y\}\subset V} d(x,y)^2$$; extremal parameter $$\mathrm{exco}_2(n, H) = \max \{\mathrm{co}_2(G) : G \text{ is } n\text{-vertex}\, H\text{-free 3-graph}\}$$.

Applying flag algebra methods, asymptotic values are obtained: $$\mathrm{exco}_2(n, K_4^3) = \frac{1}{3}\frac{n^4}{2} + o(n^4), \qquad \mathrm{exco}_2(n, K_5^3) = \frac{5}{8}\frac{n^4}{2} + o(n^4)$$ Stability results show that near extremal $K_4^3$-free hypergraphs in $\ell_2$-norm are essentially modifications of $C_n$ (Balogh et al., 2021).

5. Structural Stability and Comparison to Classical Extremals

A key property established for the $\ell_2$ extremal problem is the uniqueness (modulo $o(n^3)$ perturbations) of the $C_n$ construction in maximizing $\mathrm{co}_2$ among $K_4^3$-free hypergraphs. For the classical $\ell_1$-norm (edge count), extremal families are known to be exponentially numerous (including the Brown–Kostochka–Fon-der-Flaass constructions); however, for the $\ell_2$-norm, the structure is rigid: the extremal graphs are close to $C_n$ (the balanced complete tripartite plus one-step blow-up) (Balogh et al., 2021).

Table: Comparison of Extremal Constructions

Problem	Extremal Configuration	Uniqueness
Turán's original ($\ell_1$-norm)	Many (Brown–Kostochka etc.)	Not unique
$\ell_2$-norm (codegree-square)	$C_n$ (balanced tripartite + blow-up)	Essentially unique

This structural rigidity supports the conjecture that $C_n$ is also extremal for the classical Turán $\ell_1$-norm problem.

6. Open Problems and Perspectives

Despite advances, the original Turán (3,4)-conjecture remains unresolved. Notable directions and open problems include:

• Proving or disproving the (3,4)-domination conjecture for hypergraphs using exterior algebraic shifting.
• Characterizing equality cases in the involution-based graph theorem.
• Extending domination and shifting techniques to higher-uniformity Turán-type extremal problems and further group actions.
• Leveraging computational evidence and algebraic shifting for finer classification of extremal structures (Kalai et al., 2018).

Partial evidence for the conjecture includes success in graph analogues, computer verification for small instances (e.g., Kostochka’s examples), and the stability and uniqueness yielded in $\ell_2$-norm analyses (Balogh et al., 2021).

7. Impact and Future Directions

Algebraic, combinatorial, and analytic developments around the Tetrahedron Conjecture of Turán have provided new insights into extremal combinatorics, including toolsets based on exterior algebraic shifting, homological invariants, and flag algebra techniques. These advances refine our understanding of the interplay between structure and extremality in hypergraphs and offer pathways to resolving longstanding open problems and generalizing to broader Turán-type contexts. The persistent openness of Turán’s (3,4)-conjecture continues to stimulate methodological innovation and cross-pollination between algebraic and combinatorial extremal theory (Kalai et al., 2018, Balogh et al., 2021).