By Carlos J. Moreno
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Additional resources for Advanced Analytic Number Theory, Part I: Ramification Theoretic Methods
For each i € /, let S( be a subset of a set S. A necessary and sufficient condition for the existence of distinct representatives x,-, i = 1 , . . , w, x, € S f , x, =£ x,-, when i ^ j , is condition C: For every k = 1 , . . , Sik contain between them at least k distinct elements. Proof We have already remarked on the necessity of condition C for the existence of distinct representatives and must prove its sufficiency. We note that if each St contains only a single element x,, then condition C asserts that x x , .
We may now prove our theorem, using induction on the number n of sets, this being trivial when n = 1. ,S„}, there is a critical block Bkk not the whole system—that is, 1 ^ k < n. Deleting the elements of Bkk from the remaining sets, U consists of Bkk and a block Bn-kv, which have no elements in common. 2 condition C remains valid, and we may assume by induction that Bkk and B'„_k v both have SDR's, and being disjoint, they form together an SDR for U. Next, suppose that in the system [ / ( S j , .
N — 2 in the first row are forced. 10) n - 1 and this number is wB_2. In (c) the choices are from the array for M„-I. In (d) there is exactly one choice, namely, the third row, since only one n — 1 remains to be chosen and that is in the (n — 3)th column; similarly, then - 2,n - 3 , . . , 2 in the third row must be chosen. In (e) the choices are from the array for w„-2. 8), z„, is given by Zn = 1 + "n-2 + M„-i + 1 + U„_2 = Un + M„_2 + 2 = 0? 11) where «i» 1+^5 1 - >/5 . T «a«-2— as before. 6) may also be found recursively.
Advanced Analytic Number Theory, Part I: Ramification Theoretic Methods by Carlos J. Moreno