Speaker: Amin Bahmanian (Illinois State University)
Title: Proof of a Conjecture of Cavenagh, Hämäläinen, Lefevre, and Stones
Abstract: An r x r λ-Latin square (rectangle, respectively) is an r x s array in which each cell contains a multiset of λ elements from the set {1, ... ,r} of symbols such that each symbol occurs exactly λ times (at most λ times, respectively) in each row and column.
Cavenagh, Hämäläinen, Lefevre, and Stones asked for conditions that ensure a simple λ-Latin rectangle can be extended to a simple λ-Latin square. We solve this problem in a more general setting by allowing the number of occurrences of each symbol to be prescribed.
Cavenagh et al. also conjectured that for each r, λ there exists some n(r, λ) such that for any n ≥ n(r, k), every simple partial λ-Latin square of order r embeds in a simple λ-Latin square of order n. We confirm this conjecture.