Intersection theorems for families of matchings of complete $k$-partite $k$-graphs

TL;DR

This work extends the Erdős-Ko-Rado framework to families of $r$-matchings in the complete $k$-partite $k$-graph $\mathcal{K}(n_1,\ldots,n_k)$, linking to generalised permutations. It introduces a projection-based inductive method using $R_j$ and $S_j^i$ along with the compatibility operation $X \ltimes_i Y$ to reduce to lower-dimensional subproblems, enabling a clean bound and structural characterization. The main contributions include an Erdős-Ko-Rado type bound for intersecting families, with equality precisely for stars, and the extension to $t$-intersecting and non-uniform (set of sizes $R$) variants. These results generalise known EKR-type theorems for permutations and hypergraph matchings and provide a unified inductive approach for higher-dimensional combinatorial structures, with implications for related extremal problems and threshold phenomena.

Abstract

The celebrated {Erdős-Ko-Rado} Theorem states that for $n \geq 2k$ a family $\mathscr{F}$ of $k$ subsets of $[n]$ for which each pair of members of $\mathscr{F}$ have a non-empty intersection has size at most $\binom{n-1}{k-1}$ and for $n >2k$ has exactly this size if and only if it is the family of all $k$-subsets of $[n]$ containing a fixed element $x\in [n]$. Since its discovery, the {Erdős-Ko-Rado} Theorem has be generalised extensively and many variants have been found for structures other than sets. One such variant is for permutations and so-called generalised permutations. These structures are equivalent to $r$-matchings of the complete bipartite graph $K_{n,m}$ with $r \leq \min\{n,m\}$ in a natural way. The culmination of results of several groups of authors constitute an {Erdős-Ko-Rado} Theorem for families of generalised permutations and so for families of $r$-matchings of $K_{n,m}$ for all feasible values of $r,n$ and $m$. In this paper we generalise this by proving an {Erdős-Ko-Rado} Theorem for families of $r$-matchings of complete $k$-partite $k$-graphs, which can be seen as a partial generalisation of the {Erdős-Ko-Rado} Theorem itself. We also prove similar results for $t$-intersecting families, and for families of matchings whose members have sizes from some set of integers $R$, rather than a single size $r$.

Figures (1)

• Figure 1: A matching $M$ (in black) and its associated compatible matchings $R_4(M)$ (in blue) and $S_4^3(M)$ (in red).

Theorems & Definitions (19)

• Theorem 1.1: The Erdős-Ko-Rado Theorem MR0140419
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['thm: Gen int perms']}