Applications of Baranyai's Theorem in Information Theory

In order to motivate Baranyai's Theorem, let us present an example. Think of the director of a round{ robin tournament (for instance, a football pool or a chess tournament) who has to schedule the rounds of the tournament in such a way that i) each team plays against each other team exactly once, ii) in every round all the teams are occupied, i. e. they have an opponent. Of course, the second requirement can only be ful-lled, if there is an even number of teams or players, which we assume from now on. The following table for a tournament with 8 players obviously does the job. Here (i,j) means that Player i is scheduled against Player j. The games of the respective rounds are listed in the diierent rows. The conditions i) and ii) can be formulated in a mathematical language as follows: All possible ? 2n 2 pairs have to be distributed among 2n ? 1 rows (the rounds) such that every row partitions the set f1; : : : ; ng into pairs. Since the pairs can be regarded as the edges of the complete graph on 2n vertices, we also call this a fac-torization of the complete graph. A factor of a graph (V; E) corresponds to one round in the tournament table (i. e., each vertex occurs in exactly one edge of the factor) and a graph is factorizable if the set of edges E is the union of disjoint factors. Let us brieey analyze the above tournament table, where 2n = 8. In every round l = 1; : : : ; 2n ? 1 there are n ? 1 pairs (i; j) with i + j = l + 1 mod 2n ? 1 and one pair (2n; l+1 2 mod 2n). Such a construction is possible for all even numbers. The underlying idea is quite simple. However, the organizers of chess tournaments in the 19th century found the tables by heuristical methods. If one does so, one may run into diiculties to group the pairs of the nal rounds. The chess tournament in N urnberg 1883 almost couldn't begin in time, since the author of the tournament table, Schallopp, left the schedule at home. With the above scheme in mind it would have been no diiculty to nd the pairs of the next round. By the way, the table we presented is much …