We consider two natural generalizations of the notion of transversal to a finite hypergraph, arising in data-mining and machine learning, the so called multiple and partial transversals. We show that for an arbitrary hypergraph the families of multiple and partial transversals are both dual-bounded in the sense that the size of the corresponding dual hypergraph is bounded by a polynomial in the...

Let F be a family of convex sets in Rn and let Tm(F) be the space of m-transversals to F as subspace of the Grassmannian manifold. The purpose of this paper is to study the topology of Tm(F) through the polyhedron of configurations of (r + 1) points in Rn. This configuration space has a natural polyhedral structure with faces corresponding to what has been called order types. In particular, if ...

Given a binary quasigroup $G$ of order $n$, $d$-iterated $G[d]$ is the $(d+1)$-ary equal to $d$-times composition with itself. The Cayley table every $d$-ary $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as coincide those corresponding hypercube.
 We prove that if group $n$ satisfies Hall–Paige condition, then number transversals $ \frac...

Subnormal subgroups possessing connected transversals are briefly discussed.

A $[k,n,1]$-graph is a $k$-partite graph with parts of order $n$ such that the bipartite induced by any pair matching. An independent transversal in an set intersects each part single vertex. factor transversals pairwise-disjoint transversals. Let $f(k)$ be smallest integer $n_0$ every has assuming $n \ge n_0$. Several known conjectures imply for $k 2$, $f(k)=k$ if $k$ even and $f(k)=k+1$ odd. ...

We examine the circumstances under which certain analytic equivalence relations on Polish spaces have definable transversals.