Concerning a certain set of arrangements

  title={Concerning a certain set of arrangements},
  author={Ben Dushnik},
there exists an arrangement in S in which ak follows all the a, with i < k. Such sets S surely exist; for example, any set of m arrangements whose terminal elements are 1, 2, , m, respectively, will obviously be k-suitable for any k ? m. The smallest cardinal number N such that there exists a set of N arrangements which is k-suitable for m will be denoted by N(m, k); this is therefore a well defined positive integer for any m and k, k <m. In any collection of arrangements of the first m… 
On the Order Dimension of 1-Sets versus k-Sets
This work answers a question of Trotter by showing that dim(l,logn;n) = Θ(log3 n/log log n) and uses upper and lower bounds on f(ϑ,s,t) to derive new lower and upper bounds on dim( l,k;n).
Pairwise Suitable Family of Permutations and Boxicity
A family F of permutations of the vertices of a hypergraph H is called "pairwise suitable" for H if, for every pair of disjoint edges in H, there exists a permutation in F in which all the vertices
Constructions and nonexistence results for suitable sets of permutations
Minimal scrambling sets of simple orders
Abstract : Let 2 < or = k < n be fixed integers, a family F of simple orders on an n element set is said to be k-suitable if of every k elements in the n set, each one is the largest of the k in some
The dimension of interior levels of the Boolean lattice
AbstractLetP(k,r;n) denote the containment order generated by thek-element andr-element subsets of ann-element set, and letd(k,r;n) be its dimension. Previous research in this area has focused on the
On the λ-Dimension and the A-Dimension of Partially Ordered Sets
Let S be an abstract set and K a collection of linear orders defined on S. Define a partial order P on S as follows. For any two elements p and q in S put p < q in P if and only if p < q for each
Embedding finite posets in cubes
Dimensions of hypergraphs
Three Problems Involving Permutations
We study three problems involving permutations: the n-card problem, inv-Wilf-equivalence, and suitable sets of permutations. The n-card problem is to determine the minimal intervals [u, v] such that
Sequence Covering Arrays and Linear Extensions
A post-optimization method to make a permutation redundant so that it can be removed and the set size reduced is developed, and preliminary results on sequence covering arrays show that it is surprisingly effective.