Forbidding matching as trace in uniform hypergraphs

Abstract: We say a hypergraph $\mathcal{H}$ contains a hypergraph $\mathcal{G}$ as trace if there exists a vertex subset $S \subseteq V(\mathcal{H})$ such that $|S| = |V(\mathcal{G})|$ and ${e \cap S: e \in E(\mathcal{H})}$ contains $\mathcal{G}$ as a sub-hypergraph. We use $\mathrm{ex}r(n, \mathrm{Tr}_r(\mathcal{G}))$ to denote the maximum number of hyperedges in an $r$-uniform hypergraph on $n$ vertices not containing $\mathcal{G}$ as a trace. The study of Turán numbers for traces was initiated by Mubayi and Zhao who studied the case when $\mathcal{G}$ is a complete graph. Let $M{s+1}$ denote the graph of a matching with $s+1$ edges. In this paper, we give the upper bound of $\mathrm{ex}r(n, \mathrm{Tr}_r(M{s+1}))$ which is sharp asymptotically. When $r=3$, we give the exact value of $\mathrm{ex}3 (n, \mathrm{Tr}_3 (M{s+1}))$. We also consider the generalized Turán number in the case of matching. That is, the maximum number of copies of clique $\mathcal{K}t^r$ in hypergraphs forbidding $\mathrm{Tr}_r (M{s+1})$ as a trace. We give an upper bound which is sharp asymptotically and when $r=3$, we give the exact value. The Turán number of forbidding a matching and the other graph is another well studied topic initiated by Alon and Frankl. We also consider an analogue problem for the trace version, i.e., forbidding trace of matching and trace of complete graph as subgraphs.

• The paper provides asymptotic bounds and exact results for r-uniform hypergraphs that exclude a matching as a trace, notably determining ex_r(n, Tr_r(Mₛ₊₁)).
• It introduces a construction based on an (r-1)-uniform extremal core combined with a blow-up process, which is key to characterizing optimal structures.
• The work bridges classical Turán theory and domination parameters, extending extremal results to trace constraints and offering insights for clique maximization problems.

Extremal Problems for Traces of Matchings in Uniform Hypergraphs

Introduction and Background

The paper investigates Turán-type extremal problems in uniform hypergraphs, focusing on configurations forbidding a matching as a trace. Traces generalize induced substructures in hypergraphs: a hypergraph $\mathcal{H}$ contains a hypergraph $\mathcal{G}$ as a trace if there exists a vertex subset $S$ such that the system of intersections $\{e \cap S : e \in E(\mathcal{H})\}$ contains $\mathcal{G}$ as a subhypergraph. The function $\mathrm{ex}_r(n, \mathrm{Tr}_r(\mathcal{G}))$ denotes the maximal size of an $r$-uniform hypergraph on $n$ vertices not containing $\mathcal{G}$ as a trace.

The investigation is motivated by classical Turán-type extremal results and their extensions to hypergraphs. Previous works have resolved numerous cases for forbidding traces of (expanded) complete graphs, paths, and cycles, but the trace version for matchings remained open aside from light asymptotics. This work closes significant gaps by asymptotically (and in some cases exactly) determining the extremal functions for forbidding trace matchings, yielding new insights into the interplay between domination, clique structure, and extremal combinatorics in hypergraphs.

Main Theorems

The paper's principal results are a hierarchy of asymptotic and exact bounds for $\mathrm{ex}_r(n, \mathrm{Tr}_r(M_{s+1}))$, where $\mathcal{G}$0 is the $\mathcal{G}$1-edge matching as a forbidden trace, as well as corresponding bounds for the generalized Turán problem (maximizing cliques) and for simultaneous forbidding of different traces.

1. Asymptotics and Structure for General Uniformity

For fixed $\mathcal{G}$2 and $\mathcal{G}$3, the paper proves that for large $\mathcal{G}$4,

$\mathcal{G}$5

where $\mathcal{G}$6 is the maximum number of edges in a $\mathcal{G}$7-uniform hypergraph with dominated number at most $\mathcal{G}$8 (i.e., the maximum size for hypergraphs "resistant" to being dominated by small sets).

The main constructions combine a small extremal "core" hypergraph and a blow-up process:

• Take a $\mathcal{G}$9-uniform extremal hypergraph $S$0 with $S$1 and no isolated vertices.
• For each additional vertex $S$2, adjoin all hyperedges $S$3 for each $S$4.

This construction achieves the bound, and structural arguments show that essentially all extremal examples must follow this template (up to $S$5 exceptions).

2. Exact Results for Triple Systems ($S$6)

For $S$7, exact extremal numbers are determined: $S$8 for sufficiently large $S$9, with uniqueness characterization: the extremal hypergraphs are constructed by removing a minimum edge cover from $\{e \cap S : e \in E(\mathcal{H})\}$0 to form a core $\{e \cap S : e \in E(\mathcal{H})\}$1, blowing up as above, and adding all possible triples inside the core.

Figure 1: The dotted lines are the missing edges in $\{e \cap S : e \in E(\mathcal{H})\}$2.

3. Maximizing Clique Count Subject to Trace Constraints

For the generalized problem (maximizing the number of copies of $\{e \cap S : e \in E(\mathcal{H})\}$3), asymptotically sharp bounds are established: $\{e \cap S : e \in E(\mathcal{H})\}$4 where $\{e \cap S : e \in E(\mathcal{H})\}$5 is the maximal number of $\{e \cap S : e \in E(\mathcal{H})\}$6-uniform cliques of size $\{e \cap S : e \in E(\mathcal{H})\}$7 in a hypergraph with dominated number $\{e \cap S : e \in E(\mathcal{H})\}$8.

For $\{e \cap S : e \in E(\mathcal{H})\}$9, $\mathcal{G}$0, with constructions analogous to the edge extremal case.

4. Simultaneous Forbidding of Trace Matching and Another Trace

The extremal number forbidding both a matching and a general graph $\mathcal{G}$1 as traces is shown to be

$\mathcal{G}$2

where $\mathcal{G}$3 is the maximum number of hyperedges in an $\mathcal{G}$4-uniform hypergraph with dominated number at most $\mathcal{G}$5 and no dominated copy of $\mathcal{G}$6 for any independent set $\mathcal{G}$7 of $\mathcal{G}$8. This draws a close parallel with classic Turán-type theorems for graphs with forbidden matching number, adapted to the trace framework in hypergraphs.

Analysis and Structural Insights

A crucial technical aspect is the use of various domination and "dominated set" parameters in hypergraphs to relate the existence of large trace-free hypergraphs to known bounds and constructions in extremal graph theory (e.g., Vizing's theorem, the Erdős matching conjecture, and Kruskal-Katona).

Key structural lemmas classify hyperedges into "light" and "heavy" according to how many times their shadows can be extended, and the proofs leverage detailed analysis of trace cores and their interactions with extremal configurations. The extremum is always achieved by a small dense core and the "star-like" blowing-up construction, with the structure of the core tightly constrained by domination requirements.

Implications and Future Directions

The results establish new sharp bounds and characterizations for Turán-type extremal functions in the trace setting for hypergraph matchings, complementing previous results for Berge and expansion versions. The methods demonstrate that domination-based parameters are intrinsically linked to trace extremal problems.

The implications are both theoretical (extending the landscape of Turán extremal combinatorics in a trace context) and methodological (showing domination and enclaveless-set techniques as powerful in high-uniformity hypergraphs). The results also connect to related generalized Turán problems (maximizing clique counts, simultaneous forbidden substructures).

Two explicit conjectures are posed:

• The extremum for $\mathcal{G}$9 is conjectured to be achieved either by complete hypergraphs or by deleting exactly enough edges such that each $\mathrm{ex}_r(n, \mathrm{Tr}_r(\mathcal{G}))$0-set is covered once (when a suitable Steiner system exists).
• For the clique maximization parameter $\mathrm{ex}_r(n, \mathrm{Tr}_r(\mathcal{G}))$1, it is conjectured that the extremum is always realized by the complete $\mathrm{ex}_r(n, \mathrm{Tr}_r(\mathcal{G}))$2-uniform hypergraph on $\mathrm{ex}_r(n, \mathrm{Tr}_r(\mathcal{G}))$3 vertices.

Resolution of these conjectures would not only refine the constant terms in the main results but also clarify the structural theory of extremal trace-free hypergraphs. The paper's approach promises further development in quantitative and stability results for trace-based extremal problems, potentially informing algorithms for extremal constructions, stability analysis, and related metric parameters.

Conclusion

This work determines, up to an explicit asymptotic and often exact value, the maximal size and clique count of $\mathrm{ex}_r(n, \mathrm{Tr}_r(\mathcal{G}))$4-uniform hypergraphs forbidding a matching as a trace. The results link these extremal quantities to hypergraph domination theory and unveil structural characterizations, yielding a comprehensive picture for this class of trace problems. The conjectures and methodology motivate ongoing research into domination-restricted extremal problems and the subtle differences between trace, Berge, and expansion subgraph constraints.