On Universal Cycles for new Classes of Combinatorial Structures

A universal cycle (u-cycle) is a compact listing of a collection of combinatorial objects. In this paper, we use natural encodings of these objects to show the existence of u-cycles for collections of subsets, matroids, restricted multisets, chains of subsets, multichains, and lattice paths. For subsets, we show that a u-cycle exists for the $k$-subsets of an $n$-set if we let $k$ vary in a non zero length interval. We use this result to construct a "covering" of length $(1+o(1))$$n \choose k$ for all subsets of $[n]$ of size exactly $k$ with a specific formula for the $o(1)$ term. We also show that u-cycles exist for all $n$-length words over some alphabet $\Sigma,$ which contain all characters from $R \subset \Sigma.$ Using this result we provide u-cycles for encodings of Sperner families of size 2 and proper chains of subsets

tt-Covering Arrays Generated by a Tiling Probability Model

A t-\a covering array is an $m\times n$ matrix, with entries from an alphabet of size $\alpha$, such that for any choice of $t$ rows, and any ordered string of $t$ letters of the alphabet, there exists a column such that the "values" of the rows in that column match those of the string of letters. We use the Lov\'asz Local Lemma in conjunction with a new tiling-based probability model to improve the upper bound on the smallest number of columns $N=N(m,t,\alpha)$ of a t-\a covering array.Comment: 7 page

Shattering Thresholds for Random Systems of Sets, Words, and Permutations

This paper considers a problem that relates to the theories of covering arrays, permutation patterns, Vapnik-Chervonenkis (VC) classes, and probability thresholds. Specifically, we want to find the number of subsets of [n]:={1,2,....,n} we need to randomly select, in a certain probability space, so as to respectively "shatter" all t-subsets of [n]. Moving from subsets to words, we ask for the number of n-letter words on a q-letter alphabet that are needed to shatter all t-subwords of the q^n words of length n. Finally, we explore the number of random permutations of [n] needed to shatter (specializing to t=3), all length 3 permutation patterns in specified positions. We uncover a very sharp zero-one probability threshold for the emergence of such shattering; Talagrand's isoperimetric inequality in product spaces is used as a key tool.Comment: 25 page

Probabilistic Extensions of the Erd\H os-Ko-Rado Property

The classical Erd\H os-Ko-Rado (EKR) Theorem states that if we choose a family of subsets, each of size (k), from a fixed set of size (n (n > 2k)), then the largest possible pairwise intersecting family has size (t ={n-1\choose k-1}). We consider the probability that a randomly selected family of size (t=t_n) has the EKR property (pairwise nonempty intersection) as $n$ and $k=k_n$ tend to infinity, the latter at a specific rate. As $t$ gets large, the EKR property is less likely to occur, while as $t$ gets smaller, the EKR property is satisfied with high probability. We derive the threshold value for $t$ using Janson's inequality. Using the Stein-Chen method we show that the distribution of $X_0$, defined as the number of disjoint pairs of subsets in our family, can be approximated by a Poisson distribution. We extend our results to yield similar conclusions for $X_i$, the number of pairs of subsets that overlap in exactly $i$ elements. Finally, we show that the joint distribution $(X_0, X_1, ..., X_b)$ can be approximated by a multidimensional Poisson vector with independent components.Comment: 18 page