On the average rank of LYM-sets Original Research Article

Let S be a finite set with some rank function r such that the Whitney numbers wi = |{x ∈ S|r(x) = i}| are log-concave. Given k, N so that wk − 1 < wk ⩽ wk + m, set W = wk + wk + 1 + … + wk + m. Generalizing a theorem of Kleitman and Milner, we prove that every F ⊆ S with cardinality |F| ⩾ W has average rank at least kwk + … + (k + m) wk + m/W, provided the normalized profile vector x1, …, xn of F satisfies the following LYM-type inequality: x0 + x1 + … + xn ⩽ m + 1.