On the maximum density of $r$-graphs in which every $(r+1)$-set spans $0$ or $2$ edges

Abstract: In 1984, Frankl and Füredi asked for the maximum density of an $n$-vertex $r$-graph in which every $(r+1)$-set of vertices spans $0$ or $2$ edges. They gave a construction with asymptotic density $2<sup>{1-r}$. We significantly improve this bound by constructing such $r$-graphs with density $Ω(r<sup>{-3})$, thereby improving the dependence on $r$ from exponential to polynomial. We also obtain lower bounds for the more general problem in which every $(r+1)$-set spans an even number of edges from ${0,2,\ldots,2k}$.

• The paper introduces a new construction achieving a polynomial lower bound of Ω(r⁻³) for r-graph densities under the 0/2 edge constraint.
• It employs combinatorial techniques from coding theory and upper shadow operations, along with balanced blow-ups, to build larger graphs.
• The results bridge local edge restrictions with Turán-type extremal problems, offering new insights into hypergraph density optimization.

Maximum Density of $r$-Graphs Under Even Local Constraints: Improved Bounds

Problem Context and Prior Work

The paper "On the maximum density of $r$-graphs in which every $(r+1)$-set spans $0$ or $2$ edges" (2606.20367) addresses a foundational extremal combinatorics problem first posed by Frankl and Füredi in 1984: determining the maximum edge density of an $r$-uniform hypergraph (an $r$-graph) with the property that every $(r+1)$-subset of the vertex set spans either $0$ or $2$ edges. For $r$0, this constraint forces the graph to be complete bipartite, while for higher uniformities, the structure is substantially more complex.

Frankl and Füredi provided a geometric construction yielding asymptotic density $r$1, representing an exponential decay with $r$2. The best known upper bound follows from a double-counting argument, giving $r$3, and for specific small values (e.g., $r$4), exact or improved constructions exist. Recent advances have linked this problem to Turán-type extremal hypergraph results, particularly concerning the forbidden $r$5 configuration (three $r$6-edges on $r$7 vertices), yielding lower and upper bounds on Turán densities.

Main Results and Construction Techniques

The authors present a new construction that yields a polynomial lower bound for the density of such $r$8-graphs, $r$9, which significantly improves upon the classical exponential bound. The argument leverages combinatorial constructions rooted in coding theory and finite fields, along with the careful use of upper shadow operations and blow-ups. Key lemmas formalize how families of $(r+1)$0-edges with restricted local behavior can be transformed into $(r+1)$1-graphs with the desired $(r+1)$2 local edge counts. Notably, only the shadow operation and balanced blow-ups are employed, showing that systematic expansion preserves the local even constraint.

The main theorem states:

For every uniformity $(r+1)$3 and population size $(r+1)$4, there exists an $(r+1)$5-vertex $(r+1)$6-graph with every $(r+1)$7-set spanning $(r+1)$8 or $(r+1)$9 edges and at least $0$0 edges.

This removes the exponential dependency on $0$1 present in prior constructions, replacing it with a cubic polynomial, and does not attempt to optimize the specific constant $0$2.

The proof employs:

• Construction of $0$3-graphs with each $0$4-set containing at most one edge, via constant weight codes (Graham-Sloane bound) or finite field methods.
• Application of the upper shadow operation to generate $0$5-graphs with the requisite $0$6 property.
• Use of blow-ups and averaging down methods to achieve arbitrarily large graphs while maintaining local constraints and density.

Generalization to Even-Edge Constraints

The methodology extends to the case where every $0$7-set can span an even number of edges from $0$8 (for any fixed $0$9). The authors provide a construction yielding a further improved density, exploiting recent advances in sparse hypergraph colorings [clemen2026applications]. The key is the reduction to a problem on finding independent sets in linear hypergraphs (via the theorem of Duke, Lefmann, and Rödl), allowing for logarithmic enhancements:

For every fixed integer $2$0, there exists $2$1 such that for large $2$2, $2$3 and for $2$4, specifically, $2$5

Implications for Turán-Type Problems

This research has direct implications for Turán extremal questions: the $2$6 edge constraint is intimately linked to the classical problem of maximizing the density in $2$7-free $2$8-graphs. Improvements in maximum density under the $2$9 local rule yield improved lower bounds on Turán densities for forbidden $r$0 subgraphs. The authors note that stronger upper bounds for $r$1 graphs would translate to improved upper bounds for Turán densities via supersaturation, and conversely, recent lower bound techniques for Turán densities (e.g., tournament switching, sparse coloring) have been leveraged here for density constructions.

Concluding Remarks and Open Directions

The results demonstrate that the structure of permitted local edge counts (i.e., the set $r$2 of allowed values) is critical in determining achievable density orders. There remain fundamental open directions, particularly identifying which sets $r$3 admit density of order $r$4 or greater, versus those (including $r$5) that are restricted to the $r$6 scale.

On the practical side, these combinatorial constructions inform design principles for dense hypergraphs with forbidden local patterns—a key consideration in coding theory, network design, and extremal set systems. Theoretically, the work exposes a rich interplay between local combinatorial constraints and global density, linking design theory, coding bounds, and hypergraph Turán extremal combinatorics.

Conclusion

This paper delivers a polynomial improvement in the lower bound for the maximum density of $r$7-graphs guaranteeing every $r$8-set spans $r$9 or $r$0 edges, moving from exponential dependence to cubic in $r$1. The approach generalizes to broader local even-edge constraints via advanced sparse hypergraph coloring techniques. Theoretical advances here provide new insights into extremal hypergraph theory and Turán-type densities, suggesting further investigation into set-dependent constraints and their asymptotic density. The blend of coding-theoretic and combinatorial constructions is expected to influence forthcoming extremal combinatorics and hypergraph theory research.