Poset boxicity of graphs

AbstractA t-box representation of a graph encodes each vertex as a box in t-space determined by the (integer) coordinates of its lower and upper corner, such that vertices are adjacent if and only if the corresponding boxes intersect. The boxicity of a graph G is the minimum t for which this can be done; equivalently, it is the minimum t such that G can be expressed as the intersection graph of intervals in the t-dimensional poset that is the product of t chains. Scheinerman defined the poset boxicity of a graph G to be the minimum t such that G is the intersection graph of intervals in some t-dimensional poset. In this paper, a special class of posets is used to show that the poset boxicity of a graph on n points is at most O(log log n). Furthermore, Ramsey's theorem is used to show the existence of graphs with arbitrarily large poset boxicity

Dimensions of hypergraphs

AbstractThe dimension D(S) of a family S of subsets of n = {1, 2, …, n} is defined as the minimum number of permutations of n such that every A ∈ S is an intersection of initial segments of the permutations. Equivalent characterizations of D(S) are given in terms of suitable arrangements, interval dimension, order dimension, and the chromatic number of an associated hypergraph. We also comment on the maximum-sized family of k-element subsets of n having dimension m, and on the dimension of the family of all k-element subsets of n. The paper concludes with a series of alternative characterizations of D(S) = 2 and a list of open problems