Alon, N. The Shannon capacity of a union. Combinatorica 18, 3 (1998), 301-310.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[3, 4, 5] and Yao s [252] influential papers. Since then Ramsey theory has been applied in many di#erent ways in theoretical computer science and these have not been put together so far. Most of these applications are using existing theorems, but there are also papers, mostly by Alon [6, 7, 8], where new Ramsey type theorems are proved in the same time as an application problem is solved. Originally only results in theoretical computer science were to be mentioned but it is di#cult to separate these from relations to mathematics. In fact, the diversity of mathematical formulations for ....

...., so that among any three of them two are orthogonal, is #(n ) 6 Information Theory, Dual Source Codes It is interesting to see how various Ramsey type results are applied to information theory and communication complexity. Let G = V, E) be a graph corresponding to an information channel [8], where V represents the input set, all possible letters the channel can transmit. In each channel use a sender transmits an input and a receiver receives an output. The vertices corresponding to two letters are adjacent if and only if they can be confused, both can result in the same output. The ....

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N. Alon, The Shannon Capacity of a union, Combinatorica 18(3) (1998), 301--310.

....bound on #(G) This is, however, not better that #: Proposition 5.6 Suppose that G has an orthonormal representation in dimension d. Then #(G) # d. On the other hand, if we consider orthogonal representations over fields of finite characteristic, the dimension may be a better bound than # [37, 5]. This, however, goes outside the ideas of semidefinite optimization. To relate # to the Shannon capacity of a graph, the following is the key observation: 20 Proposition 5.7 For any two graphs, #(G H) #(G)#(H) and #(G H) #(G)#(H) It is now easy to generalize the bound for the ....

....#(G H) #(G)#(H) It is now easy to generalize the bound for the Shannon capacity of the pentagon, given in the introduction, to arbitrary graphs. Corollary 5. 8 For every graph, #(G) # #(G) Does equality hold here Examples by Haemers [37] and more recent much sharper examples by Alon [5] show that the answer is negative in general. But we can derive at least one interesting class of examples from the general results below. Proposition 5.9 For every graph G, #(G)#(G) # n. If G has a vertex transitive automorphism group, then equality holds. Corollary 5.10 If G is a ....

N. Alon, The Shannon capacity of a union, Combinatorica 18 (1998), 301--310.

....constructible low rank co diagonal matrices over Z 6 imply explicit Ramsey graph constructions. Our best construction reproduces the logarithmic order of magnitude of the Ramsey graph of Frankl and Wilson [5] continuing the sequence of results on new explicit Ramsey graph constructions of Alon [1] and Grolmusz [6] Our present result, analogously to the constructions of [6] and [1] can be generalized to more than one color. 1 the electronic journal of combinatorics 7 (2000) #R15 2 Our results give a recipe for constructing explicit Ramsey graphs from explicit low rank co diagonal ....

.... Our best construction reproduces the logarithmic order of magnitude of the Ramsey graph of Frankl and Wilson [5] continuing the sequence of results on new explicit Ramsey graph constructions of Alon [1] and Grolmusz [6] Our present result, analogously to the constructions of [6] and [1], can be generalized to more than one color. 1 the electronic journal of combinatorics 7 (2000) #R15 2 Our results give a recipe for constructing explicit Ramsey graphs from explicit low rank co diagonal matrices over Z 6 , analogously to the way that our results gave a method for constructing ....

N. Alon. The Shannon capacity of a union. Combinatorica, 18:301--310, 1998.

....= V; E) is the maximal size of a set of disjoint sets from E . Thus the matching number equals the packing number, in which the underlying hypergraph really is a graph and the disjoint sets of E are the independent edges. At the end of this historical sketch we return to the Shannon capacity. In [4], Alon disproved a conjecture of the work mentioned already, 13] Shannon proved in [13] that for every two graphs G and H the inequality Theta(G [H) Theta(G) Theta(H) holds and that equality holds for graphs with special properties. Shannon remarked in [13] We conjecture but have not ....

....results. Section 3 contains the calculation of the matching numbers of n th powers of K l;m [K p;q and the matching capacity. Moreover we compare the results to the Shannon capacity and the structure of the vertex independence number of the n th powers of the Pentagon. Also Alon s counterexample [4] is presented here. Section 4 treats the special incomplete case (K l;m nK c;d ) K p;q and we present the matching numbers of the n powers. Section 5 may be seen as an appendix: Brualdi s basic bounds [5] are presented and tested in our situation. I want to thank Prof. Ahlswede, who lead me to ....

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N. Alon, The Shannon capacity of a union, Combinatorica 18, 301-310, 1998.

....a considerable amount of efforts by various researchers, and the best known explicit construction is due to Frankl and Wilson [9] who gave an explicit 2 edge coloring of the complete graph on m (1 o(1) log m 4 log log m vertices with no monochromatic clique on m vertices. See also [11] [2], for some multi colored variations. These constructions do not supply any nontrivial explicit lower bounds for R(s; m) where s is fixed and m grows. Such constructions for s = 3 appear in various papers, see [1] 6] where it is shown that R(3; m) Omega Gamma m 3=2 ) via an explicit ....

N. Alon, The Shannon Capacity of a union, Combinatorica 18 (1998), 301-310.

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Alon, N. The Shannon capacity of a union. Combinatorica 18, 3 (1998), 301-310.

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N. Alon, The Shannon Capacity of a union, Combinatorica 18 (1998), 301-310.

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N. Alon, The Shannon capacity of a union, Combinatorica, 18 (1998), 301-310.

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