On large sets of Kirkman systems with holes

Further results on large sets of Kirkman triple systems

LR design is introduced by the second author in his recent paper, and it plays a very important role in the construction of LKTS (a large set of disjoint Kirkman triple system). In this paper, we generalize it and introduce a new design RPICS. Some constructions for these two designs are also presented. With the relationship between them and LKTS, we obtain some new LKTSs.

A construction for large sets of disjoint Kirkman triple systems

An improved product construction for large sets of Kirkman triple systems

It has been shown by Lei, in his recent paper, that there exists a large set of Kirkman triple systems of order uv (LKTS(uv)) if there exist an LKTS(v), a TKTS(v) and an LR(u), where a TKTS(v) is a transitive Kirkman triple system of order v, and an LR(u) is a new kind of design introduced by Lei. In this paper, we improve this product construction by removing the condition “there exists a TKTS...

On Large Systems of Sets with No Large Weak ∆-subsystems

The notion of a weak ∆-system was introduced and studied by Erdős, Milner and Rado [5] in 1974. A weak ∆-system is a family of sets where all pairs of sets have the same intersection size. Erdős and Szemerédi [7] investigated the behavior of the function F (n,r)—the largest integer so that there exists a family F of subsets of an n-element set which does not contain a ∆-system of r sets. Answer...

Large convex holes in random point sets

A convex hole (or empty convex polygon) of a point set P in the plane is a convex polygon with vertices in P , containing no points of P in its interior. Let R be a bounded convex region in the plane. We show that the expected number of vertices of the largest convex hole of a set of n random points chosen independently and uniformly over R is Θ(logn/(log log n)), regardless of the shape of R.