Power-Saving Upper Bound in 3-Uniform Hypergraphs

• Power-Saving Upper Bound is a result in extremal hypergraph theory that demonstrates how forbidding local dense configurations forces global sparsity.
• The approach uses iterative hypergraph constructions, deficiency analysis, and sunflower reductions to systematically lower the edge density from quadratic order.
• The result refines previous bounds by proving that, for every e ≥ 3, an additive buffer of ⌊log₂ e⌋+38 guarantees a reduction to O(n^(2-ε)), marking a genuine power-saving improvement.

A power-saving upper bound in the context of the Brown–Erdős–Sós problem refers to a nontrivial reduction in the maximal edge density achievable in a 3-uniform hypergraph on $n$ vertices, while avoiding specific local dense configurations. The concept formalizes the extent to which local forbidden structures enforce global sparsity beyond what is implied by standard density arguments, measuring "power-saving" as a strict improvement over quadratic dependence in $n$.

1. Definitions and Notations

The function $f(n,v,e)$ is defined as the maximal number of edges in a 3-uniform hypergraph on $n$ vertices that avoids any subhypergraph with $v$ vertices spanning at least $e$ edges (i.e., avoids every $(v,e)$-configuration). Here, a 3-uniform hypergraph $H$ consists of a vertex set $V$ and edge set $E$, where each edge is a set of exactly three vertices. The paper focuses on how dense $H$ can be under the constraint that no "local" subset of $v$ vertices supports the desired number $e$ of edges.

2. Historical Context and Previous Results

The Brown–Erdős–Sós problem, posed in 1973, questioned whether $f(n, e+3, e) = o(n^2)$ holds for all $e \geq 3$; equivalently, whether prohibiting every occurrence of $e$ edges on $e+3$ vertices strictly subquadratically bounds the total edge count. The case $e=3$ is the (6,3)-theorem of Ruzsa–Szemerédi. Sárközy and Selkow (2004) established that $f(n, e + \lfloor \log_2 e \rfloor + 2, e) = o(n^2)$ is true for every $e \geq 3$. Later work by Conlon, Gishboliner, Levanzov, and Shapira refined this, showing that the additive constant needed can be further lowered, and investigated variants with even smaller buffer terms.

3. New Contributions

The major advance of the cited paper is a quantitative power-saving improvement. Specifically, for every $e \geq 3$, there exists $\varepsilon > 0$ such that

$$f(n, e + \lfloor \log_2 e \rfloor + 38, e) = O(n^{2-\varepsilon})$$

where the exponent $2-\varepsilon$ signals a "power-saving" reduction below quadratic order. This achievement is very near the Sárközy–Selkow bound but gains a genuine reduction in the exponent by allowing a small additive increase (the constant $38$). The significance lies in obtaining stronger control of global density—forcing sparsity—by admitting only a slightly larger local forbidden configuration.

4. Conjectures and Open Problems

The original BES conjecture posits that $f(n, e+3, e) = o(n^2)$ for every $e \geq 3$. Gowers and Long conjectured even stronger: $f(n, e+4, e) = O(n^{2-\varepsilon})$ for some $\varepsilon=\varepsilon(e)>0$. The function $d(e)$, the minimal buffer ensuring a power-saving upper bound,

$f(n, e + d(e), e) = O(n^{2-\varepsilon})$

remains a focal point, with the current best proven result being $d(e) = \lfloor \log_2 e \rfloor + 38$. The challenge is to minimize $d(e)$ (possibly down to 4), thereby tightening the link between local forbidden structures and global sparsity.

5. Mathematical Formulations and Proof Elements

The proof constructs a sequence of "eligible" hypergraphs $F_0, F_1, ..., F_\ell$ with

$$e(F_j) = 2^j \cdot e(F_0), \quad \Delta(F_j) = \Delta(F_0) + j,$$

where $\Delta(F) = v(F) - e(F)$ is the deficiency. The approach utilizes "good" independent sets $A \subset V(F)$ with the property that any $U \supset A$ has deficiency at least $|A|+1$, and builds upon structural lemmas establishing control over deficiency when combining constructions. Additionally, the concept of an $(r,F)$-sunflower is employed—multiple copies of $F$ intersecting at a core $U$ (with $\Delta(U) \geq \Delta(F)$)—facilitating cleanup or reduction steps to obtain sharp configuration control.

The main result, formally: $$f(n, e + \lfloor \log_2 e \rfloor + 38, e) = O(n^{2-\varepsilon(e)}),$$ with explicit $\varepsilon(e)>0$, is a power-saving upper bound near the previously known thresholds.

6. Applications and Implications

The Brown–Erdős–Sós-type problems shape extremal combinatorics and indirectly inform results in additive number theory (e.g., connections to Roth's theorem) and Turán-type questions in hypergraphs. Establishing power-saving bounds strengthens arguments about unavoidable local structure causing global sparsity and unlocks sharper density theorems in related areas. More generally, the iterative, deficiency-controlled hypergraph constructions and sunflower reduction mechanisms developed here may be adapted to other combinatorial settings requiring local-to-global transition analysis.

7. Conclusion and Future Directions

The demonstrated result closes the gap to the Sárközy–Selkow bound for 3-uniform hypergraphs, delivering a quantitative power-saving improvement at the cost of a small additive constant. Remaining open are whether the constant $38$ or the logarithmic buffer can be eliminated or reduced to approach $d(e)=4$, aligning with the Gowers–Long conjecture. Achieving such results would mark yet stronger links between local constraint and global sparse structure, with significant consequences for extremal hypergraph theory. The techniques developed—eligible hypergraph sequences, deficiency analysis, and sunflower reductions—are likely to impact future work addressing optimal upper bounds where local exclusion implies global sparsity.