Tournaments and Score Sets by Chuck Buchwald, Sierra College Math.
In 1978, Dr. Reid conjectured that every finite non-empty set S of non-negative integers is the score set for some tournament. In fact, he proved this when |S| = 1, 2, 3. Later in 1984, M. Hager showed the claim was true when |S| = 4, 5 and finally in 1987 Yao Tian-xing proved the conjecture for any positive integer |S| by an arithmetical argument. In other words, given a set S = {a1, a2, ..., an}, we know there exists a tournament in which every vertex has a score chosen from this set. Furthermore we know that every integer from this set is the score of some vertex in this tournament. Our goal is to obtain constructions for these tournaments when |S| = 2 by combining the results of Dr. Reid and one of the many constructive proofs of Landau's Theorem . We will visually demonstrate our results with a program written in Matlab that will generate these tournaments.