Upper bounds on the general covering number C_λ(v, k, t, m)

Abstract

A collection C of k-subsets (called blocks) of a v v-set Xðv vÞ ¼ f1; 2; . . . ; v vg (with elements called points) is called a t-ðv v; k; m; Þ covering if for every m-subset M of Xðv vÞ there is a subcollection K of C with jKj ! such that every block K 2 K has at least t points in common with M. It is required that v v ! k ! t and v v ! m ! t. The minimum number of blocks in a t-ðv v; k; m; Þ covering is denoted by C ðv v; k; t; mÞ. We present some constructions producing the best known upper bounds on C ðv v; k; t; mÞ for k ¼ 6, a parameter of interest to lottery players.

Linear spaces with small generated subspaces

“…Let L ⊂ Z be the line through x, y. It suffices to show that the vertices of the component of G containing (z, 1) or (z, 2) belong to (Z \ L) × {1, 2}. First, observe that vertices from L belong to cycles fully contained in L × {1, 2}.…”

“…Often, K = {k} has been considered in previous work on this topic. See [1] for a recent survey of results and connections to lottery designs.…”

Asymptotic Bounds for General Covering Designs

“…For t = 1, the limit d(k, 1, p) = p [17] trivially follows from (6). In the following sections, we will therefore assume that t > 1.…”

“…In Section 4, a construction for general covering designs that works for arbitrary values of n, k, t, p, and λ will be described. This construction will then lead to general upper bounds on δ(k, t, p) for arbitrary values of k, t, and p. In the remaining sections, we will present combinatorial constructions that result in improvements of the best-known upper bounds for δ (5,4,5) (new bound 2.002, old bound 2.037), for δ (5,4,6) (new bound 3.667, old bound 3.768) and for δ (4,4,6) (new bound 2.633, old bound 2.649). The last new bound is equivalent to a new lower bound for the Turán density π(K 4 6 ), which will be defined in Section 5.1.…”

“…The last new bound is equivalent to a new lower bound for the Turán density π(K 4 6 ), which will be defined in Section 5.1. Novel bounds on δ (7,4,5) and δ (7,4,6) will be introduced in Section 5.4 by means of a class of objects called partitionable Steiner systems. Finally, some new upper bounds for C(n, k, t, p) when n is small will also be provided.…”