L. Gargano, J. Korner and U. Vaccaro, Capacities: from information theory to extremal set theory, Journal of Combinatorial Theory A, Vol. 68 (1994), pp. 296--316.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a recently discovered extension of the concept of Shannon capacity of a graph. Denote by L(n) the maximum cardinality of a family of pairs (A i , B i ) of subsets of an n set satisfying B A j B j (i j ) B j =#if and only if i=j. Then L(n) n asymptotically in n. This is proved in [9]. For small values of i and a quadratic ground field we obtain improvements by means of Hermitian codes. Consider the Hermitian curve defined by the equation X Y Z 0 over the field F q 2 of constants. This curve has genus and q 1 rational points. These form the well known ....

L. Gargano, J. Korner and U. Vaccaro, Capacities: from information theory to extremal set theory, Journal of Combinatorial Theory A, Vol. 68 (1994), pp. 296--316.

....cf. Calderbank, Frankl, Graham, Li and Shepp [4] and Blokhuis [3] This does not mean, however, that this generalization is an academic exercise of no immediate use. To the contrary, Sperner capacity became the key to the solution of some important open problems in extremal set theory, 8] [9]. The study of the extension of a poset to sequences of its elements leads to the equally justified notion of antichain capacity that we will introduce in this paper. Although our concept will be defined for arbitrary oriented graphs, not just those corresponding to posets, it is useful to present ....

....we should mention that the always existing limit Sigma(G) lim n 1 1 n log N(G;n) is called the Sperner capacity of the oriented graph G. Notice that the definition can be extended to any directed graph and thus Sperner capacity is a formal generalization of Shannon capacity, cf. [9]. Now, two sequences, x and y in V n are called G independent if x i and y i are equal or non adjacent vertices of G for any i, 1 i n. Finally, the sequences x and y in V n are called G unrelated if they are either G incomparable or G independent. Let M(G;n) denote the largest ....

L. Gargano, J. Korner, U. Vaccaro, Capacities: from information theory to extremal set theory, J. Comb. Theory, Ser. A, to appear

....and information theory often deal with the same problem from a different perspective. The two fields hardly ever touch. Still, in the sixties, Alfr ed R enyi s inspiring work was showing how fruitful the connections between these two areas can be for both. Recently, in a series of papers ( 9] [10], 15] or, for the earlier work: 5] and [17] we have applied a novel information theoretic construction method combined with traditional entropy based bounding techniques to determine the precise exponential asymptotics in a large number of combinatorial problems of which the most outstanding ....

....contain an other. All of these problems can be reformulated in such a way that the maximum size of a set of n length sequences from a fixed alphabet is to be determined under various restrictions on the element pairs present in the same coordinate of any two sequences from the set. In [9] [10] and [15] one requires the presence of some (directed) edges of certain (directed) graphs between the coordinates. Such problems include Sperner s theorem and its various generalizations, the Shannon capacity of a graph, R enyi s question on qualitatively independent partitions (posed as a problem ....

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L. Gargano, J. Korner, U. Vaccaro, "Capacities: from information theory to extremal set theory", J. Comb. Theory, Ser. A, to appear,

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L. Gargano, J. Korner, and U. Vaccaro. Capacities: from information theory to extremal set theory. J. Combinatorial Theory, Series A. To appear.

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