The k-th power of a graph G is the graph whose vertex set is V(G) k , where two distinct k-tuples are adjacent iff they are equal or adjacent in G in each coordinate. The Shannon capacity of G, c(G), is lim k→∞ α(G k )1/k , where α(G) denotes the independence number of G. When G is the characteristic graph of a channel C, c(G) measures the effective alphabet size of C in a zero-error protocol. A sum of channels, C = ∑ i C i , describes a setting when there are t ≥ 2 senders, each with his own channel C i , and each letter in a word can be selected from any of the channels. This corresponds to a disjoint union of the characteristic graphs, G = ∑ i G i . It is well known that c(G) ≥ ∑ i c(G i ), and in  it is shown that in fact c(G) can be larger than any fixed power of the above sum. We extend the ideas of  and show that for every F, a family of subsets of [t], it is possible to assign a channel C i to each sender i [t], such that the capacity of a group of senders X ⊂ [t] is high iff X contains some F F. This corresponds to a case where only privileged subsets of senders are allowed to transmit in a high rate. For instance, as an analogue to secret sharing, it is possible to ensure that whenever at least k senders combine their channels, they obtain a high capacity, however every group of k - 1 senders has a low capacity (and yet is not totally denied of service). In the process, we obtain an explicit Ramsey construction of an edge-coloring of the complete graph on n vertices by t colors, where every induced subgraph on exp (Ω (√log n\log \log n}) vertices contains all t colors.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Computational Mathematics