The Intersection Problem and Different Pairs Problem for Latin Squares

摘要

The intersection of two Latin squares of the same order is the set of cells that contain the same entries in both Latin squares. Determining the order of this set can be asked for any type of Latin square and has been solved for most. Generalizing this to Latin squares of different orders leads to a conjecture of Dukes and Mendelsohn, which will be shown to be true. Results on the intersection of Latin squares, idempotent Latin squares, and idempotent symmetric Latin squares are explored. The relationship between the intersection problem for Latin squares and the intersection problem for Steiner triple systems will also be investigated. In addition to new results, past results are included presenting a common and clear notation. The proofs of some new results are able to replace proofs of past results as well as present a straightforward proof structure to new and past results.;Two Latin squares of the same order are said to be r-orthogonal if the set of pairs occurring in corresponding cells has size r. Using this notation, two orthogonal Latin squares of order n are n2-orthogonal. The idea of r-orthogonality is generalized to Latin squares of different orders. The set of possible values is established for r and it is shown that this possible set can be obtained for pairs of Latin squares with certain orders.