By Ainouche A., Schiermeyer I.

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An administrative council is composed of a set X of individuals. Each of them carries a certain weight in decisions, and it is required that every set E C X carrying a total weight greater than some threshold fixed in advance, should have access t o documents kept in a safe with multiple locks. The minimal “coalitions” which can open the safe constitute a simple hypergraph H . The problem consists in determining the number of locks necessary so that by giving one or more keys to every individual, the safe can be opened if and only if at least one of the coalitions of H is present.

Note that if G is a path P, we obtain Helly’s Theorem). T h e o r e m 11 (Tuza [1984]). ,Em)be a s i m p l e k-Helly hypergraph of order n. If min Bj 2 k+1 then I J (*) Proof. 1. We shall show first that every edge E j contains a vertex a, such that Ei-{uj} is not contained in any edge other than E j . ,r. ,1’. ,ETis non-empty. -1 and since H is k-Helly, we have also: r n E~ # 0. j- 1 A contradiction follows. 2 . Thus every edge E, contains a vertex a, such that (Ej-{a,}) n (X-Ei) # Set Ej = Ej-{aj} F; = X-Ej.

The theorem holds trivially for r = 1 or m = 1; proceed now by induction on r and on m. + (,",) which is what we had to show. Suppose now that As a consequence we can write From ( l ) ,and applying the induction hypothesis on m to N - H ( x , ) , which contradicts (4). Corollary. Let H be an r-uniform hypergraph and let k be an integer with r > k 2 2. If a is the largest integer such that m ( H ) 2 (f) then 4 W I k ) 1);( Proof. Let H I be a partial hypergraph of H with m ( H l ) = (:). Q2)9 m(~2Ir-2) etc.