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Let ∆ n−1 denote the (n − 1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of ∆ n−1 obtained by starting with the full 1-dimensional skeleton of ∆ n−1 and then adding each 2−simplex independently with probability p. Let H 1 (Y ; F 2) denote the first ho-mology group of Y with mod 2 coefficients. It is shown that for any function ω(n) that(More)
Abstruct-A shared directional multichannel (SDM) consists of a set of inputs and a set of outputs to which we connect transmitters and receivers, respectively. A signal placed at any given input reaches a subset of the outputs, and a channel is specified by the sets of outputs that are reachable from each input. A message is received successfully at an(More)
Intersection and measured intersection graphs are quite common in the literature. In this paper we introduce the analogous concept of measured difference graphs: Given an arbitrary hypergraph H = {H1, ..., Hn}, let us associate to it a graph on vertex set [n] = {1, 2, ..., n} in which (i, j) is an edge iff the corresponding sets Hi and Hj are " sufficiently(More)