Notes on Craven's conjecture.

Craven's conjecture says that 2 sup (n-1) is the maximum number of linear orders on {1,2,...,n} that simultaneously satisfy certain restrictions on three-element subsets of {1,2,...,n}. This is true for n = 3, but the maximum exceeds 2 sup (n-1) for n >= 4. There is a set of nine linear orders on {1,2,3,4} that satisfy the restrictions. As n gets large, the ratio of the size of the maximum satisfying set to 2 sup (n-1) approaches infinity.