105 research outputs found
A Sidon-type condition on set systems
Consider families of -subsets (or blocks) on a ground set of size .
Recall that if all -subsets occur with the same frequency , one
obtains a -design with index . On the other hand, if all
-subsets occur with different frequencies, such a family has been called (by
Sarvate and others) a -adesign. An elementary observation shows that such
families always exist for . Here, we study the smallest possible
maximum frequency .
The exact value of is noted for and an upper bound (best possible
up to a constant multiple) is obtained for using PBD closure. Weaker, yet
still reasonable asymptotic bounds on for higher follow from a
probabilistic argument. Some connections are made with the famous Sidon problem
of additive number theory.Comment: 6 page
The linear system for Sudoku and a fractional completion threshold
We study a system of linear equations associated with Sudoku latin squares.
The coefficient matrix of the normal system has various symmetries arising
from Sudoku. From this, we find the eigenvalues and eigenvectors of , and
compute a generalized inverse. Then, using linear perturbation methods, we
obtain a fractional completion guarantee for sufficiently large and sparse
rectangular-box Sudoku puzzles
- …