D-interval hypergraph

In graph theory, a $d$-interval hypergraph is a kind of a hypergraph constructed using intervals of real lines. The parameter $d$ is a positive integer. The vertices of a $d$-interval hypergraph are the points of $d$ disjoint lines (thus there are uncountably many vertices). The edges of the graph are $d$-tuples of intervals, one interval in every real line.

The simplest case is $d = 1$. The vertex set of a 1-interval hypergraph is the set of real numbers; each edge in such a hypergraph is an interval of the real line. For example, the set ${ [−2, −1], [0, 5], [3, 7] }$ defines a 1-interval hypergraph. Note the difference from an interval graph: in an interval graph, the vertices are the intervals (a finite set); in a 1-interval hypergraph, the vertices are all points in the real line (an uncountable set).

As another example, in a 2-interval hypergraph, the vertex set is the disjoint union of two real lines, and each edge is a union of two intervals: one in line #1 and one in line #2.

The following two concepts are defined for $d$-interval hypergraphs just like for finite hypergraphs:


 * A matching is a set of non-intersecting edges, i.e., a set of non-intersecting $d$-intervals. For example, in the 1-interval hypergraph ${ [−2, −1], [0, 5], [3, 7] },$ the set ${ [−2, −1], [0, 5] }$ is a matching of size 2, but the set ${ [0, 5], [3, 7] }$ is not a matching since its elements intersect. The largest matching size in $H$ is denoted by $ν(H)$.
 * A covering or a transversal is a set of vertices that intersects every edge, i.e., a set of points that intersects every $d$-interval. For example, in the 1-interval hypergraph ${ [−2, −1], [0, 5], [3, 7] },$ the set ${−1.5, 4}$ is a covering of size 2, but the set ${−1.5, 1}$ is not a covering since it does not intersect the edge $[3, 7]$. The smallest transversal size in $H$ is denoted by $τ(H)$.

$ν(H) ≤ τ(H)$ is true for any hypergraph $H$.

Tibor Gallai proved that, in a 1-interval hypergraph, they are equal: $τ(H) = ν(H)$. This is analogous to Kőnig's theorem for bipartite graphs.

Gabor Tardos proved that, in a 2-interval hypergraph, $τ(H) ≤ 2ν(H)$, and it is tight (i.e., every 2-interval hypergraph with a matching of size $m$, can be covered by $2m$ points).

Kaiser proved that, in a $d$-interval hypergraph, $τ(H) ≤ d(d – 1)ν(H)$, and moreover, every $d$-interval hypergraph with a matching of size $m$, can be covered by at $d(d − 1)m$ points, $(d − 1)m$ points on each line.

Frick and Zerbib proved a colorful ("rainbow") version of this theorem.