Berge-Mₖ Turán Numbers

• Berge-Mₖ is defined as an r-uniform hypergraph with a bijection from a matching of k independent edges to its hyperedges.
• The Turán number exᵣ(n, Berge-Mₖ) represents the maximum number of edges in an n-vertex r-uniform hypergraph that avoids containing a Berge-Mₖ.
• Recent research establishes sharp phase transitions and extremal constructions using degree analysis, greedy matching, and combinatorial techniques.

A Berge-$M_k$ is an $r$-uniform hypergraph admitting a bijection $\phi$ from the edge set of a matching $M_k$ (a graph of $k$ independent edges) to its hyperedges, with each graph edge contained in its corresponding hyperedge. The Turán number $\ex_r(n,\mathrm{Berge}{-}M_k)$ is the maximal cardinality of an $n$-vertex $r$-uniform hypergraph with no Berge-$M_k$ as a subhypergraph. Recent research provides precise formulas, extremal configurations, and generalizations for these Turán numbers across hypergraph regimes, establishing a rich phase structure for their asymptotics and sharpness (Palmer et al., 2017, Zhao et al., 6 Oct 2025, Zhou et al., 19 Jun 2025).

1. Formal Definition and Background

For fixed $k\geq1$ and $r$0, the interest lies in $r$1, the largest number of edges in an $r$2-vertex $r$3-uniform hypergraph $r$4 avoiding a Berge-$r$5. A matching $r$6 consists of $r$7 independent edges—even in the hypergraph setting, the notion of "vertex-disjoint edges" is realized via the Berge mapping, so that $r$8 hyperedges correspond to $r$9 graph edges with pairwise-disjoint vertex sets.

This investigation is motivated by the broader theory of Turán-type extremal problems for forbidden subhypergraphs, where the transition from classic graph matchings to their Berge analogues in uniform hypergraphs introduces subtle partitioning phenomena, dependence on the uniformity parameter $\phi$0, and intricate combinatorial constructions. The Berge-$\phi$1 formulation generalizes classic graph forbidden subgraph problems and connects to expansion techniques, core-set methods, degree sequences, and greedy matching protocols.

2. Piecewise Exact Formula for $\phi$2

Let $\phi$3, $\phi$4, and $\phi$5, for some threshold $\phi$6. The precise Turán number is given by the following piecewise formula (Palmer et al., 2017, Zhou et al., 19 Jun 2025, Zhao et al., 6 Oct 2025) (with $\phi$7 notation used for $\phi$8):

$\phi$9

For $M_k$0 (the "star regime"), the leading term is $M_k$1 with lower-order correction $M_k$2. For $M_k$3, the extremal configuration consists of all $M_k$4-sets containing a fixed $M_k$5-set, yielding $M_k$6 edges. For $M_k$7, the complete $M_k$8-hypergraph on $M_k$9 vertices achieves the bound. When $k$0, the bound is the constant $k$1, as no $k$2 vertex-disjoint $k$3-sets exist.

3. Extremal Constructions and Attaining Sharpness

The sharpness of these formulas is certified by explicit extremal hypergraph constructions, depending on the regime:

• Star Construction ($k$4):

Fix $k$5 with $k$6. All $k$7-sets meeting $k$8 form the extremal family. Any $k$9 hyperedges must share a vertex from $\ex_r(n,\mathrm{Berge}{-}M_k)$0, so no $\ex_r(n,\mathrm{Berge}{-}M_k)$1 are disjoint. Counting yields the formula above.

• Clique Construction ($\ex_r(n,\mathrm{Berge}{-}M_k)$2):

The complete $\ex_r(n,\mathrm{Berge}{-}M_k)$3-uniform hypergraph on $\ex_r(n,\mathrm{Berge}{-}M_k)$4 vertices: all $\ex_r(n,\mathrm{Berge}{-}M_k)$5 possible edges. As $\ex_r(n,\mathrm{Berge}{-}M_k)$6, any $\ex_r(n,\mathrm{Berge}{-}M_k)$7 edges can not be mutually disjoint.

• $\ex_r(n,\mathrm{Berge}{-}M_k)$8 Regime:

All $\ex_r(n,\mathrm{Berge}{-}M_k)$9-sets containing a fixed $n$0-set constitute the unique extremal configuration, yielding $n$1 edges.

• Sparse Regime ($n$2):

In this case, only $n$3 mutually disjoint $n$4-edges can exist, so any $n$5 hyperedges form the extremal system.

Every construction achieves the stated bound with equality, verifying exactness for all parameter choices.

4. Proof Schemes and Key Combinatorial Principles

Upper bounds are shown via two principal arguments:

• Degree Analysis and Core-Set Reduction:

If a vertex in the hypergraph has degree above a threshold, a Berge-star structure emerges, facilitating seed formation for a larger Berge-matching through reduction to a link hypergraph. Inductive arguments on the matching number then force the appearance of a forbidden Berge-$n$6.

• Greedy Matching Argument:

If all degrees are lower than the threshold, a greedy procedure selects disjoint hyperedges, incrementally building toward a Berge-matching. Exceeding the exact bound guarantees a collection of disjoint edges violating $n$7-freeness.

This analysis is applied separately to the star, clique, and sparse regimes, exploiting the combinatorial structure and uniformity of the hypergraph in each case. Lower bounds are confirmed by explicit construction as described above.

5. Phase Transition and Asymptotics

A pronounced phase transition structure is present in the asymptotic behavior of $n$8 as $n$9 (Palmer et al., 2017, Zhao et al., 6 Oct 2025):

Regime	Growth Order	Leading Term
$r$0	Linear	$r$1
$r$2	Linear	$r$3
$r$4	Constant (in $r$5)	$r$6
$r$7	Constant (in $r$8)	$r$9

This sharp transition, from linear to constant, is dictated by whether sufficiently many vertices exist to assemble large matchings, with extremal bounds collapsing when $M_k$0 exceeds the threshold for disjointness.

6. Generalizations: Berge Matchings With Additional Constraints

Recent expansions handle Turán numbers for families forbidding Berge-matchings jointly with other substructures:

• Berge-$M_k$1 and a Family $M_k$2:

For $M_k$3 and a suitable $M_k$4-graph $M_k$5, if either $M_k$6 or $M_k$7 lacks certain strong colorings:

$M_k$8

where $M_k$9 comprises all $k\geq1$0-graphs from $k\geq1$1 by deleting strongly independent sets (Zhao et al., 6 Oct 2025).

• Berge Matching Plus Berge Bipartite Graphs:

If $k\geq1$2 is bipartite with vertex-cover number $k\geq1$3, then for $k\geq1$4,

$k\geq1$5

Additional regimes are classified based on $k\geq1$6 and $k\geq1$7.

• General Upper Bound Lemma:

For Berge-families $k\geq1$8,

$k\geq1$9

Where edge colorings partition $r$00 for optimizing derived Turán bounds.

These generalizations leverage advanced arguments, including Gallai–Edmonds decompositions, link graphs, colorings, and shadow techniques to control combinatorial proliferation in the presence of multiple forbidden families (Zhao et al., 6 Oct 2025).

7. Connections to Broader Turán-Type Theory and Open Directions

The determination of Turán numbers for Berge-$r$01 bridges classic graph-theoretic extremal questions—such as Erdős–Ko–Rado, Erdős–Frankl–Füredi, and shadows of set systems—to uniform hypergraph settings, emphasizing phase transitions, sharpness, and structure of extremal families. Extensions to forbidding more complex structures, such as bipartite Berge graphs or joint matching-and-graph conditions, highlight a direction toward broader family-based Turán extremal problems, with implications for coloring, covering, and decomposition methods (Palmer et al., 2017, Zhao et al., 6 Oct 2025).

The study of exact formulas and sharpness in intermediate regimes, as well as for small values of $r$02, remains an active area, and the structural connections between hypergraph expansion and forbidden Berge configurations continue to animate ongoing extremal combinatorial research.