Ahlswede–Khachatrian Theorems: Weighted, Infinite, and Hamming
نویسنده
چکیده
The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of a k-uniform t-intersecting family on n points, and describes all optimal families. We extend this theorem to several other settings: the weighted case, the case of infinitely many points, and the Hamming scheme. The weighted Ahlswede–Khachatrian theorem gives the maximal μp measure of a t-intersecting family on n points, where μp(A) = p |A|(1−p)n−|A|. As has been observed by Ahlswede and Khachatrian and by Dinur and Safra, this theorem can be derived from the classical one by a simple reduction. However, this reduction fails to identify the optimal families, and only works for p < 1/2. We translate the two original proofs of Ahlswede and Khachatrian to the weighted case, thus identifying the optimal families in all cases. We also extend the theorem to the case p > 1/2, using a different technique of Ahlswede and Khachatrian (the case p = 1/2 is Katona’s intersection theorem). We then extend the weighted Ahlswede–Khachatrian theorem to the case of infinitely many points. The Ahlswede–Khachatrian theorem on the Hamming scheme gives the maximum cardinality of a subset of Zm in which any two elements x, y have t positions i1, . . . , it such that xij − yij ∈ {−(s − 1), . . . , s − 1}. We show that this case corresponds to μp with p = s/m, extending work of Ahlswede and Khachatrian, who considered the case s = 1. We also determine the maximum cardinality families. We obtain similar results for subsets of [0, 1], though in this case we are not able to identify all maximum cardinality families.
منابع مشابه
The Edge-Diametric Theorem in Hamming Spaces
The maximum number of edges spanned by a subset of given diameter in a Hamming space with alphabet size at least three is determined. The binary case was solved earlier by Ahlswede and Khachatrian [A diametric theorem for edges, J. Combin. Theory Ser. A 92(1) (2000) 1–16]. © 2007 Elsevier B.V. All rights reserved.
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