Let k, d, λ > 1 be integers with d > λ and let X ⊂ R be a finite set. A (d−λ)-plane L transversal to the convex hull of all k-sets of X is called Kneser transversal. If in addition L contains (d− λ) + 1 points of X, then L is called complete Kneser transversal. In this paper, we present various results on the existence of (complete) Kneser transversals for λ = 2, 3. In order to do this, we intr...

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 the hypergraphs of all multiple and all partial transversals are dualbounded in the sense that in both cases, the size of the dual hypergraph is bounded by a polynomial in the cardinality and the l...

We generalize the Hadwiger theorem on line transversals to collections of compact convex sets in the plane to the case where the sets are connected and the transversals form an arrangement of pseudolines. The proof uses the embeddability of pseudoline arrangements in topological affine planes. In 1940 Santaló showed [8], by an example, that Vincensini’s proof [9] of an extension of Helly’s theo...

A Latin square of order n is an n × n array of n symbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of n entries, one selected from each row and each column of a Latin square of order n such that no two entries contain the same symbol. Define T (n) to be the maximum number of transversals over all Latin squares of order n. We show that bn ≤ T (n) ≤ ...

Day and Kasperczyk (1985) and Day, Watson, and Jolly (1986) claimed that the Poggendorff illusion, in a Tolansky-type vertical bisection task, is no smaller when the line transversals are replaced with dots and the oblique parallels are removed, despite other evidence that transversals and parallels contribute significantly to the illusion. Wenderoth, O'Connor, and Johnson (1986) showed indirec...

The classic Coupon-Collector Problem (CCP) is generalized. Only basic probability theory is used. Centerpiece rather is an algorithm that efficiently counts all k-element transversals of a set system.