Actually it turned out later that Avi was just quoting Tali Tishby’s joke, and this Joke have led Tali to his early work on BosonSampling.

]]>The sixth formula is from the paper titled “The Shannon Capacity of a union” in Combinatorica 18 (3) 1998. The formula disproves a conjecture of Shannon, namely that the Shannon capacity is subadditive with respect to the disjoint union of graphs. In the paper it is proved that there is a graph G such that both G and its complement has Shannon capacity at most k, and the formula holds. Thus the disjoint union of two graphs can have superpolinomially larger capacity than the original graphs. The argument uses explicit construction, the graph is a Kneser-style graph with 3 parameters:G(r,s,p). The vertices are subsets of size s from [r], and two such subsets are connected iff their intersection is of size -1 modulo p. The proof that the graph satisfies the formula involves both number theory and linear algebra.

The latest independent place where the formula (not the result) itself is reproduced might be the survey “Ramsey Theory Applications” by Vera Rosta. The formula appears on the lower half of page 14. The survey is a dynamic one, and it is last modified in 2004, so there might be room for improvement, unless the author modifies it in the near future. (this fortunate event is unlikely to happen, and I do not plan to take any steps towards it :-) )

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