Appendix: On Set Coverings in Cartesian Product Spaces
نویسنده
چکیده
Consider (X, E), where X is a finite set and E is a system of subsets whose union equals X. For every natural number n ∈ N define the cartesian products Xn = ∏n 1 X and En = ∏n 1 E . The following problem is investigated: how many sets of En are needed to cover Xn? Let this number be denoted by c(n). It is proved that for all n ∈ N exp{C · n} ≤ c(n) ≤ exp{Cn+ log n+ log log |X|}+ 1. A formula for C is given. The result generalizes to the case where X and E are not necessarily finite and also to the case of non–identical factors in the product. As applications one obtains estimates on the minimal size of an externally stable set in cartesian product graphs and also estimates on the minimal number of cliques needed to cover such graphs. 1 A Covering Theorem Let X be a non–empty set with finitely many elements and let E be a set of non–empty subsets of X with the property ⋃ E∈E E = X . (We do not introduce an index set for E in order to keep the notations simple). For n ∈ N, the set of natural numbers, we define the cartesian product spaces Xn = ∏n 1 X and En = ∏n 1 E . The elements of En can be viewed as subsets of Xn. We say that E ′ n ⊂ En covers Xn or is a covering of Xn, if Xn = ⋃ En∈E′ n En. We are interested in obtaining bounds on the numbers c(n) defined by c(n) = min E′ n covers Xn |E ′ n|, n ∈ N. (1) Clearly, c(n1 + n2) ≤ c(n1) · c(n2) for n1, n2 ∈ N. Example 1 below shows that equality does not hold in general. Denote by Q the set of all probability distributions on the finite set E , denote by 1E(·) the indicator function of a set E, and define K by K = max q∈Q min x∈X ∑
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تاریخ انتشار 2006