On the Density of Sets Containing No k-Element Arithmetic Progression of a Certain Kind
نویسندگان
چکیده
A theorem now known as Sperner’s Lemma [5] states that a largest collection of subsets of an n-element set such that no subset contains another is obtained by taking the collection of all the subsets with cardinal bn=2c. (We denote by bxc, resp. dxe, the largest integer less than or equal to x, resp. the smallest integer greater than or equal to x.) In other words, the density of a largest antichain in the set of all subsets of an n-element set is 2 n n bn=2c : The generalization of this problem which is considered here was mentioned to one of the authors by R. L. Graham. An antichain in an n-element set can be viewed as a collection of sequences of length n on the symbols 0;1 such that no two sequences occur which are, in some order, the rows of a 2 n matrix in which each column is either constant of is 0 1 :
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On the Density of Sets Containing No ^-element Arithmetic Progression of a Certain Kind
A theorem now known as Sperner's Lemma [5] states that a largest collection of subsets of an n-element set such that no subset contains another is obtained by taking the collection of all the subsets with cardinal [n/2\. (We denote by |*J, resp. [*], the largest integer less than or equal to x, resp. the smallest integer greater than or equal to x.) In other words, the density of a largest anti...
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